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Very early difficulties II

Very early difficulties II

This is the second part of a series of posts about certain difficulties math students have in the very early stages of studying abstract math. The first post, Very early difficulties in studying abstract math, gives some background to the subject and discusses one particular difficulty: Some students do not know that it is worthwhile to try starting a proof by rewriting what is to be proved using the definitions of the terms involved.

Math StackExchange

The website Math StackExchange is open to any questions about math, even very easy ones. It is in contrast with Math OverFlow, which is aimed at professional mathematicians asking questions in their own field.

Math SE contains many examples of the early difficulties discussed in this series of posts, and I recommend to math ed people (not just RUME people, since some abstract math occurs in advanced high school courses) that they might consider reading through questions on Math SE for examples of misunderstanding students have.

There are two caveats:

  • Most questions on Math SE are at a high enough level that they don’t really concern these early difficulties.
  • Many of the questions are so confused that it is hard to pinpoint what is causing the difficulty that the questioner has.

Connotations of English words

The terms(s) defined in a definition are often given ordinary English words as names, and the beginner automatically associates the connotations of the meaning of the English word with the objects defined in the definition.

Infinite cardinals

If $A$ if a finite set, the cardinality of $A$ is simply a natural number (including $0$). If $A$ is a proper subset of another set $B$, then the cardinality of $A$ is strictly less than the cardinality of $B$.

In the nineteenth century, mathematicians extended the definition of cardinality for infinite sets, and for the most part cardinality has the same behavior as for finite sets. For example, the cardinal numbers are well-ordered. However, for infinite sets it is possible for a set and a proper subset of the set to have the same cardinality. For example, the cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers. This phenomenon causes major cognitive dissonance.

Question 1331680 on Math Stack Exchange shows an example of this confusion. I have also discussed the problem with cardinality in the abstractmath.org section Cardinality.

Morphism in category theory

The concept of category is defined by saying there is a bunch of objects called objects (sorry bout that) and a bunch of objects called morphisms, subject to certain axioms. One requirement is that there are functions from morphisms to objects choosing a “domain” and a “codomain” of each morphism. This is spelled out in Category Theory in Wikibooks, and in any other book on category theory.

The concepts of morphism, domain and codomain in a category are therefore defined by abstract definitions, which means that any property of morphisms and their domains and codomains that is true in every category must follow from the axioms. However, the word “morphism” and the talk about domains and codomains naturally suggests to many students that a morphism must be a function, so they immediately and incorrectly expect to evaluate it at an element of its domain, or to treat it as a function in other ways.

Example

If $\mathcal{C}$ is a category, its opposite category $\mathcal{C}^{op}$ is defined this way:

  • The objects of $\mathcal{C}^{op}$ are the objects of $\mathcal{C}$.
  • A morphism $f:X\to Y$ of $\mathcal{C}^{op}$ is a morphism from $Y$ to $X$ of $\mathcal{C}$ (swap the domain and codomain).

In Question 980933 on Math SE, the questioner is saying (among other things) that in $\text{Set}^{op}$, this would imply that there has to be a morphism from a nonempty set to the empty set. This of course is true, but the questioner is worried that you can’t have a function from a nonempty set to the empty set. That is also true, but what it implies is that in $\text{Set}^{op}$, the morphism from $\{1,2,3\}$ to the empty set is not a function from $\{1,2,3\}$ to the empty set. The morphism exists, but it is not a function. This does not any any sense make the definition of $\text{Set}^{op}$ incorrect.

Student confusion like this tends to make the teacher want to have a one foot by six foot billboard in his classroom saying

A MORPHISM DOESN’T HAVE TO BE A FUNCTION!

However, even that statement causes confusion. The questioner who asked Question 1594658 essentially responded to the statement in purple prose above by assuming a morphism that is “not a function” must have two distinct values at some input!

That questioner is still allowing the connotations of the word “morphism” to lead them to assume something that the definition of category does not give: that the morphism can evaluate elements of the domain to give elements of the codomain.

So we need a more elaborate poster in the classroom:

The definition of “category” makes no requirement
that an object has elements
or that morphisms evaluate elements.

As was remarked long long ago, category theory is pointless.

English words implementing logic

There are lots of questions about logic that show that students really do not think that the definition of some particular logical construction can possibly be correct. That is why in the abstractmath.org chapter on definitions I inserted this purple prose:

A definition is a totalitarian dictator.

It is often the case that you can explain why the definition is worded the way it is, and of course when you can you should. But it is also true that the student has to grovel and obey the definition no matter how weird they think it is.

Formula and term

In logic you learn that a formula is a statement with variables in it, for example “$\exists x((x+5)^3\gt2)$”. The expression “$(x+5)^3$” is not a formula because it is not a statement; it is a “term”. But in English, $H_2O$ is a formula, the formula for water. As a result, some students have a remarkably difficult time understanding the difference between “term” and “formula”. I think that is because those students don’t really believe that the definition must be taken seriously.

Exclusive or

Question 804250 in MathSE says:

“Consider $P$ and $Q$. Let $P+Q$ denote exclusive or. Then if $P$ and $Q$ are both true or are both false then $P+Q$ is false. If one of them is true and one of them is false then $P+Q$ is true. By exclusive or I mean $P$ or $Q$ but not both. I have been trying to figure out why the truth table is the way it is. For example if $P$ is true and $Q$ is true then no matter what would it be true?”

I believe that the questioner is really confused by the plus sign: $P+Q$ ought to be true if $P$ and $Q$ are both true because that’s what the plus sign ought to mean.

Yes, I know this is about a symbol instead of an English word, but I think the difficulty has the same dynamics as the English-word examples I have given.

If I have understood this difficulty correctly, it is similar to the students who want to know why $1$ is not a prime number. In that case, there is a good explanation.

Only if

The phrase “only if” simply does not mean the same thing in math as it does in English. In Question 17562 in MathSE, a reader asks the question, why does “$P$ only if $Q$” mean the same as “if $P$ then $Q$” instead of “if $Q$ then $P$”?

Many answerers wasted a lot of time trying to convince us that “$P$ only if $Q$” mean the same as “if $P$ then $Q$” in ordinary English, when in fact it does not. That’s because in English, clauses involving “if” usually connote causation, which does not happen in math English.

Consider these two pairs of examples.

  1. “I take my umbrella only if it is raining.”
  2. “If I take my umbrella, then it is raining.”
  3. “I flip that switch only if a light comes on.”
  4. “If I flip that switch, a light comes on.”

The average non-mathematical English speaker will easily believe that (1) and (4) are true, but will balk and (2) and (3). To me, (3) means that the light coming on makes me flip the switch. (2) is more problematical, but it does (to me) have a feeling of causation going the wrong way. It is this difference that causes students to balk at the equivalence in math of “$P$ only if $Q$” and “If $P$, then $Q$”. In math, there is no such thing as causation, and the truth tables for implication force us to live with the fact that these two sentences mean the same thing.

Henning Makholm’ answer to Question 17562 begins this way: “I don’t think there’s really anything to understand here. One simply has to learn as a fact that in mathematics jargon the words ‘only if’ invariably encode that particular meaning. It is not really forced by the everyday meanings of ‘only’ and’ if’ in isolation; it’s just how it is.” That is the best way to answer the question. (Other answerers besides Makholm said something similar.)

I have also discussed this difficulty (and other difficulties with logic) in the abmath section on “only if“.

References

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Functions: Metaphors, Images and Representations

Please read this post at abstractmath.org. I originally posted the document here but some of the diagrams would not render, and I haven’t been able to figure out why. Sorry for having to redirect.

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Problems caused for students by the two languages of math

The two languages of math

Mathematics is communicated using two languages: Mathematical English and the symbolic language of math (more about them in two languages).

This post is a collection of examples of the sorts of trouble that the two languages cause beginning abstract math students. I have gathered many of them here since they are scattered throughout the literature. I would welcome suggestions for other references to problems caused by the languages of math.

In many of the examples, I give links to the literature and leave you to fish out the details there. Almost all of the links are to documents on the internet.

There is an extensive list of references.

Conjectures

Scattered through this post are conjectures. Like most of my writing about difficulties students have with math language, these conjectures are based on personal observation over 37 years of teaching mostly computer engineering and math majors. The only hard research of any sort I have done in math ed consists of the 426 citations of written mathematical writing included in the Handbook of Mathematical Discourse.

Disclaimer

This post is an attempt to gather together the ways in which math language causes trouble for students. It is even more preliminary and rough than most of my other posts.

  • The arrangement of the topics is unsatisfactory. Indeed, the topics are so interrelated that it is probably impossible to give a satisfactory linear order to them. That is where writing on line helps: Lots of forward and backward references.
  • Other people and I have written extensively about some of the topics, and they have lots of links. Other topics are stubs and need to be filled out. I have probably missed important points about and references to many of them.
  • Please note that many of the most important difficulties that students have with understanding mathematical ideas are not caused by the languages of math and are not represented here.

I expect to revise this article periodically as I find more references and examples and understand some of the topics better. Suggestions would be very welcome.

Intricate symbolic expressions

I have occasionally had students tell me that have great difficulty understanding a complicated symbolic expression. They can’t just look at it and learn something about what it means.

Example

Consider the symbolic expression \[\displaystyle\left(\frac{x^3-10}{3 e^{-x}+1}\right)^6\]

Now, I could read this expression aloud as if it were text, or more precisely describe it so that someone else could write it down. But if I am in math mode and see this expression I don’t “read” it, even to myself.

I am one of those people who much of the time think in pictures or abstractions without words. (See references here.)

In this case I would look at the expression as a structured picture. I could determine a number of things about it, and when I was explaining it I would point at the board, not try to pronounce it or part of it:

  • The denominator is always positive so the expression is defined for all reals.
  • The exponent is even so the value of the expression is always nonnegative. I would say, “This (pointing at the exponent) is an even power so the expression is never negative.”
  • It is zero in exactly one place, namely $x=\sqrt[3]{10}$.
  • Its derivative is also $0$ at $\sqrt[3]{10}$. You can see this without calculating the formula for the derivative (ugh).

There is much more about this example in Zooming and Chunking.

Algebra in high school

There are many high school students stymied by algebra, never do well at it, and hate math as a result. I have known many such people over the years. A revealing remark that I have heard many times is that “algebra is totally meaningless to me”. This is sometimes accompanied by a remark that geometry is “obvious” or something similar. This may be because they think they have to “read” an algebraic expression instead of studying it as they would a graph or a diagram.

Conjecture

Many beginning abstractmath students have difficulty understanding a symbolic expression like the one above. Could this be cause by resistance to treating the expression as a structure to be studied?

Context-sensitive pronunciation

A symbolic assertion (“formula” to logicians) can be embedded in a math English sentence in different ways, requiring the symbolic assertion to be pronounced in different ways. The assertion itself is not modified in any way in these different situations.

I used the phrase “symbolic assertion” in abstractmath.org because students are confused by the logicians’ use of “formula“.
In everyday English, “$\text{H}_2\text{O}$” is the “formula” for water, but it is a term, not an assertion.

Example

“For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$.”

  • In the sentence above, the assertion “$x\gt0$” must be pronounced “$x$ that is greater than $0$” or something similar.
  • The standalone assertion “$x\gt0$” is pronounced “$x$ is greater than $0$.”
  • The sentence “Let $x\gt0$” must be pronounced “Let $x$ be greater than $0$”.

The consequence is that the symbolic assertion, in this case “$x\gt0$”, does not reveal that role it plays in the math English sentence that it is embedded in.

Many of the examples occurring later in the post are also examples of context-sensitive pronunciation.

Conjectures

Many students are subconsciously bothered by the way the same symbolic expression is pronounced differently in different math English sentences.

This probably impedes some students’ progress. Teachers should point this phenomenon out with examples.

Students should be discouraged from pronouncing mathematical expressions.

For one thing, this could get you into trouble. Consider pronouncing “$\sqrt{3+5}+6$”. In any case, when you are reading any text you don’t pronounce the words, you just take in their meaning. Why not take in the meaning of algebraic expressions in the same way?

Parenthetic assertions

A parenthetic assertion is a symbolic assertion embedded in a sentence in math English in such a way that is a subordinate clause.

Example

In the math English sentence

“For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$”

mentioned above, the symbolic assertion “$x\gt0$” plays the role of a subordinate clause.

It is not merely that the pronunciation is different compared to that of the independent statement “$x\gt0$”. The math English sentence is hard to parse. The obvious (to an experienced mathematician) meaning is that the beginning of the sentence can be read this way: “For every real number $x$, which is bigger than $0$…”.

But new student might try to read it is “For every real number $x$ is greater than $0$ …” by literally substituting the standalone meaning of “$x\gt0$” where it occurs in the sentence. This makes the text what linguists call a garden path sentence. The student has to stop and start over to try to make sense of it, and the symbolic expression lacks the natural language hints that help understand how it should be read.

Note that the other two symbolic expressions in the sentence are not parenthetic assertions. The phrase “real number” needs to be followed by a term, and it is, and the phrase “such that” must be followed by a clause, and it is.

More examples

  • “Consider the circle $S^1\subseteq\mathbb{C}=\mathbb{R}^2$.” This has subordinate clauses to depth 2.
  • “The infinite series $\displaystyle\sum_{k=1}^\infty\frac{1}{k^2}$ converges to $\displaystyle\zeta(2)=\frac{\pi^2}{6}\approx1.65$”
  • “We define a null set in $I:=[a,b]$ to be a set that can be covered by a countable of intervals with arbitrarily small total length.” This shows a parenthetical definition.
  • “Let $F:A\to B$ be a function.”
    A type declaration is a function? In any case, it would be better to write this sentence simply as “Let $F:A\to B$”.

David Butler’s post Contrapositive grammar has other good examples.

Math texts are in general badly written. Students need to be taught how to read badly written math as well as how to write math clearly. Those that succeed (in my observation) in being able to read math texts often solve the problem by glancing at what is written and then reconstructing what the author is supposedly saying.

Conjectures

Some students are baffled, or at least bothered consciously or unconsciously, by parenthetic assertions, because the clues that would exist in a purely English statement are missing.

Nevertheless, many if not most math students read parenthetic assertions correctly the first time and never even notice how peculiar they are.

What makes the difference between them and the students who are stymied by parenthetic assertions?

There is another conjecture concerning parenthetic assertions below.

Context-sensitive meaning

“If” in definitions

Example

The word “if” in definitions does not mean the same thing that it means in other math statements.

  • In the definition “An integer is even if it is divisible by $2$,” “if” means “if and only if”. In particular, the definition implies that a function is not even if it is not divisible by $2$.
  • In a theorem, for example “If a function is differentiable, then it is continuous”, the word “if” has the usual one-way meaning. In particular, in this case, a continuous function might not be differentiable.

Context-sensitive meaning occurs in ordinary English as well. Think of a strike in baseball.

Conjectures

The nearly universal custom of using “if” to mean “if and only if” in definitions makes it a harder for students to understand implication.

This custom is not the major problem in understanding the role of definitions. See my article Definitions.

Underlying sets

Example

In a course in group theory, a lecturer may say at one point, “Let $F:G\to H$ be a homomorphism”, and at another point, “Let $g\in G$”.

In the first sentence, $G$ refers to the group, and in the second sentence it refers to the underlying set of the group.

This usage is almost universal. I think the difficulty it causes is subtle. When you refer to $\mathbb{R}$, for example, you (usually) are referring to the set of real numbers together with all its canonical structure. The way students think of it, a real number comes with its many relations and connections with the other real numbers, ordering, field properties, topology, and so on.

But in a group theory class, you may define the Klein $4$-group to be $\mathbb{Z}_2\times\mathbb{Z}_2$. Later you may say “the symmetry group of a rectangle that is not a square is the Klein $4$-group.” Almost invariably some student will balk at this.

Referring to a group by naming its underlying set is also an example of synecdoche.

Conjecture

Students expect every important set in math to have a canonical structure. When they get into a course that is a bit more abstract, suddenly the same set can have different structures, and math objects with different underlying sets can have the same structure. This catastrophic shift in a way of thinking should be described explicitly with examples.

Way back when, it got mighty upsetting when the earth started going around the sun instead of vice versa. Remind your students that these upheavals happen in the math world too.

Overloaded notation

Identity elements

A particular text may refer to the identity element of any group as $e$.

This is as far as I know not a problem for students. I think I know why: There is a generic identity element. The identity element in any group is an instantiation of that generic identity element. The generic identity element exists in the sketch for groups; every group is a functor defined on that sketch. (Or if you insist, the generic identity element exists in the first order theory for groups.) I suspect mathematicians subconsciously think of identity elements in this way.

Matrix multiplication

Matrix multiplication is not commutative. A student may forget this and write $(A^2B^2=(AB)^2$. This also happens in group theory courses.

This problem occurs because the symbolic language uses the same symbol for many different operations, in this case the juxtaposition notation for multiplication. This phenomenon is called overloaded notation and is discussed in abstractmath.org here.

Conjecture

Noncommutative binary operations written using juxtaposition cause students trouble because going to noncommutative operations requires abandoning some overlearned reflexes in doing algebra.

Identity elements seem to behave the same in any binary operation, so there are no reflexes to unlearn. There are generic binary operations of various types as well. That’s why mathematicians are comfortable overloading juxtaposition. But to get to be a mathematician you have to unlearn some reflexes.

Negation

Sometimes you need to reword a math statement that contains symbolic expressions. This particularly causes trouble in connection with negation.

Ordinary English

The English language is notorious among language learners for making it complicated to negate a sentence. The negation of “I saw that movie” is “I did not see that movie”. (You have to put “d** not” (using the appropriate form of “do”) before the verb and then modify the verb appropriately.) You can’t just say “I not saw that movie” (as in Spanish) or “I saw not that movie” (as in German).

Conjecture

The method in English used to negate a sentence may cause problems with math students whose native language is not English. (But does it cause math problems with those students?)

Negating symbolic expressions

Examples

  • The negation of “$n$ is even and a prime” is “$n$ is either odd or it is not a prime”. The negation should not be written “$n$ is not even and a prime” because that sentence is ambiguous. In the heat of doing a proof students may sometimes think the negation is “$n$ is odd and $n$ is not a prime,” essentially forgetting about DeMorgan. (He must roll over in his grave a lot.)
  • The negation of “$x\gt0$” is “$x\leq0$”. It is not “$x\lt0$”. This is a very common mistake.

These examples are difficulties caused by not understanding the math. They are not directly caused by difficulties with the languages of math.

Negating expressions containing parenthetic assertions

Suppose you want to prove:

“If $f:\mathbb{R}\to\mathbb{R}$ is differentiable, then $f$ is continuous”.

A good way to do this is by using the contrapositive. A mechanical way of writing the contrapositive is:

“If $f$ is not continuous, then $f:\mathbb{R}\to\mathbb{R}$ is not differentiable.”

That is not good. The sentence needs to be massaged:

“If $f:\mathbb{R}\to\mathbb{R}$ is not continuous, then $f$ is not differentiable.”

Even better would be to write the original sentence as:

“Suppose $f:\mathbb{R}\to\mathbb{R}$. Then if $f$ is differentiable, then $f$ is continuous.”

This is discussed in detail in David Butler’s post Contrapositive grammar.

Conjecture

Students need to be taught to understand parenthetic assertions that occur in the symbolic language and to learn to extract a parenthetic assertion and write it as a standalone assertion ahead of the statement it occurs in.

Scope

The scope of a word or variable consists of the part of the text for which its current definition is in effect.

Examples

  • “Suppose $n$ is divisible by $4$.” The scope is probably the current paragraph or perhaps the current proof. This means that the properties of $n$ are constrained in that section of the text.
  • “In this book, all rings are unitary.” This will hold for the whole book.

There are many more examples in the abstractmath.org article Scope.

If you are a grasshopper (you like to dive into the middle of a book or paper to find out what it says), knowing the scope of a variable can be hard to determine. It is particularly difficult for commonly used words or symbols that have been defined differently from the usual usage. You may not suspect that this has happened since it might be define once early in the text. Some books on writing mathematics have urged writers to keep global definitions to a minimum. This is good advice.

Finding the scope is considerably easier when the text is online and you can search for the definition.

Conjecture

Knowing the scope of a word or variable can be difficult. It is particular hard when the word or variable has a large scope (chapter or whole book.)

Variables

Variables are often introduced in math writing and then used in the subsequent discussion. In a complicated discussion, several variables may be referred to that have different statuses, some of them introduced several pages before. There are many particular ways discussed below that can cause trouble for students. This post is restricted to trouble in connection with the languages of math. The concept of variable is difficult in itself, not just because of the way the math languages represent them, but that is not covered here.

Much of this part of the post is based on work of Susanna Epp, including three papers listed in the references. Her papers also include many references to other work in the math ed literature that have to do with understanding variables.

See also Variables in abstractmath.org and Variables in Wikipedia.

Types

Students blunder by forgetting the type of the variable they are dealing with. The example given previously of problems with matrix multiplication is occasioned by forgetting the type of a variable.

Conjecture

Students sometimes have problems because they forget the data type of the variables they are dealing with. This is primarily causes by overloaded notation.

Dependent and independent

If you define $y=x^2+1$, then $x$ is an independent variable and $y$ is a dependent variable. But dependence and independence of variablesare more general than that example suggests.
In an epsilon-delta proof of the limit of a function (example below,) $\varepsilon$ is independent and $\delta$ is dependent on $\varepsilon$, although not functionally dependent.

Conjecture

Distinguishing dependent and independent variables causes problems, particularly when the dependence is not clearly functional.

I recently ran across a discussion of this on the internet but failed to record where I saw it. Help!

Bound and free

This causes trouble with integration, among other things. It is discussed in abstractmath.org in Variables and Substitution. I expect to add some references to the math ed literature soon.

Instantiation

Some of these variables may be given by existential instantiation, in which case they are dependent on variables that define them. Others may be given by universal instantiation, in which case the variable is generic; it is independent of other variables, and you can’t impose arbitrary restrictions on it.

Existential instantiation

A theorem that an object exists under certain conditions allows you to name it and use it by that name in further arguments.

Example

Suppose $m$ and $n$ are integers. Then by definition, $m$ divides $n$ if there is an integer $q$ such that $n=qm$. Then you can use “$q$” in further discussion, but $q$ depends on $m$ and $n$. You must not use it with any other meaning unless you start a new paragraph and redefine it.

So the following (start of a) “proof” blunders by ignoring this restriction:

Theorem: Prove that if an integer $m$ divides both integers $n$ and $p$, then $m$ divides $n+p$.

“Proof”: Let $n = qm$ and $p = qm$…”

Universal instantiation

It is a theorem that for any integer $n$, there is no integer strictly between $n$ and $n+1$. So if you are given an arbitrary integer $k$, there is no integer strictly between $k$ and $k+1$. There is no integer between $42$ and $43$.

By itself, universal instantiation does not seem to cause problems, provided you pay attention to the types of your variables. (“There is no integer between $\pi$ and $\pi+1$” is false.)

However, when you introduce variables using both universal and existential quantification, students can get confused.

Example

Consider the definition of limit:

Definition: $\lim_{x\to a} f(x)=L$ if and only if for every $\epsilon\gt0$ there is a $\delta\gt0$ for which if $|x-a|\lt\delta$ then $|f(x)-L|\lt\epsilon$.

A proof for a particular instance of this definition is given in detail in Rabbits out of a Hat. In this proof, you may not put constraints on $\epsilon$ except the given one that it is positive. On the other hand, you have to come up with a definition of $\delta$ and prove that it works. The $\delta$ depends on what $f$, $a$ and $L$ are, but there are always infinitely many values of $\delta$ which fit the constraints, and you have to come up with only one. So in general, two people doing this proof will not get the same answer.

Reference

Susanna Epp’s paper Proof issues with existential quantification discusses the problems that students have with both existential and universal quantification with excellent examples. In particular, that paper gives examples of problems students have that are not hinted at here.

References

A nearly final version of The Handbook of Mathematical Discourse is available on the web with links, including all the citations. This version contains some broken links. I am unable to recompile it because TeX has evolved enough since 2003 that the source no longer compiles. The paperback version (without the citations) can be bought as a book here. (There are usually cheaper used versions on Amazon.)

Abstractmath.org is a website for beginning students in abstract mathematics. It includes most of the material in the Handbook, but not the citations. The Introduction gives you a clue as to what it is about.

Two languages

My take on the two languages of math are discussed in these articles:

The Language of Mathematics, by Mohan Ganesalingam, covers these two languages in more detail than any other book I know of. He says right away on page 18 that mathematical language consists of “textual sentences with symbolic material embedded like ‘islands’ in the text.” So for him, math language is one language.

I have envisioned two separate languages for math in abstractmath.org and in the Handbook, because in fact you can in principle translate any mathematical text into either English or logical notation (first order logic or type theory), although the result in either case would be impossible to understand for any sizeable text.

Topics in abstractmath.org

Context-sensitive interpretation.

“If” in definitions.

Mathematical English.

Parenthetic assertion.

Scope

Semantic contamination.

Substitution.

The symbolic language of math

Variables.

Zooming and Chunking.

Topics in the Handbook of mathematical discourse.

These topics have a strong overlap with the topics with the same name in abstractmath.org. They are included here because the Handbook contains links to citations of the usage.

Context-sensitive.

“If” in definitions.

Parenthetic assertion.

Substitution.

Posts in Gyre&Gimble

Names

Naming mathematical objects

Rabbits out of a Hat.

Semantics of algebra I.

Syntactic and semantic thinkers

Technical meanings clash with everyday meanings

Thinking without words.

Three kinds of mathematical thinkers

Variations in meaning in math.

Other references

Contrapositive grammar, blog post by David Butler.

Proof issues with existential quantification, by Susanna Epp.

The role of logic in teaching proof, by Susanna Epp (2003).

The language of quantification in mathematics instruction, by Susanna Epp (1999).

The Language of Mathematics: A Linguistic and Philosophical Investigation
by Mohan Ganesalingam, 2013. (Not available from the internet.)

On the communication of mathematical reasoning, by Atish Bagchi, and Charles Wells (1998a), PRIMUS, volume 8, pages 15–27.

Variables in Wikipedia.

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The definition of “function”

 

This is the new version of the abstractmath article on the definition of function. I had to adapt the formatting and some of it looks weird, but legible. It is prettier on abstractmath.org.

I expect to announce new revisions of other abmath articles on this blog, with links, but not to publish them here. This article brings out a new point of view about defining functions that I wanted to call attention to, so I am publishing it here, as well.

 

FUNCTIONS: SPECIFICATION AND DEFINITION

It is essential that you understand many of the images, metaphors and terminology that mathe­maticians use when they think and talk about functions. For many purposes, the precise mathematical definition of "function" does not play much of a role when you are trying to understand particular kinds of functions. But there is one point of view about functions that has resulted in fundamental progress in math:

 

 

A function is a mathematical object.

To deal with functions in that way you need a precise definition of "function". That is what this article gives you.

  • The article starts by giving a specification of "function".
  • After that, we get into the technicalities of the definitions of the general concept of function.
  • Things get complicated because there are several inequivalent definitions of "function" in common use.

Specification of "function"

A function $f$ is a mathematical object which determines and is completely determined by the following data:

(DOM) $f$ has a domain, which is a set. The domain may be denoted by $\text{dom} f$.

(COD) $f$ has a codomain, which is also a set and may be denoted by $\text{cod} f$.

(VAL) For each element $a$ of the domain of $f$, $f$ has a value at $a$, denoted by $f(a)$.

(FP) The value of $f$ at $a$ is completely determined by $a$ and $f$.

(VIC) The value of $f$ at $a$ must be an element of the codomain of $f$.

  • The operation of finding $f(a)$ given $f$ and $a$ is called evaluation.
  • "FP" means functional property.
  • "VIC" means "value in codomain".

Examples

The examples of functions chapter contains many examples. The two I give here provide immediate examples.

A finite function

Let $F$ be the function defined on the set $\left\{1,\,2,3,6 \right\}$ as follows: $F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1$. This is the function called "Finite'' in the chapter on examples of functions.

  • The definition of $F$ says "$F$ is defined on the set $\left\{1,\,2,\,3,\,6 \right\}$". That phrase means that the domain is that set.
  • The value of $F$ at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that $F(2) = 3$. No other reason needs to be given. Mathematical definitions can be arbitrary.
  • The codomain of $F$ is not specified, but must include the set $\{1,2,3\}$. The codomain of a function is often not specified when it is not important — which is most of the time in freshman calculus (for example).

A real-valued function

Let $G$ be the real-valued function defined by the formula $G(x)={{x}^{2}}+2x+5$.

  • The definition of $G$ gives the value at each element of the domain by a formula. The value at $3$, for example, is $G(3)=3^2+2\cdot3+5=20$.
  • The definition of $G$ does not specify the domain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is $\mathbb{R}$.
  • The definition does not specify the codomain, either. However, must include all real numbers greater than or equal to 4. (Why?)

What the specification means

  • The specification guarantees that a function satisfies all five of the properties listed.
  • The specification does not define a mathematical structure in the way mathematical structures have been defined in the past: In particular, it does not require a function to be one or more sets with structure.
  • Even so, it is useful to have the specification, because:

     

     

    Many mathematical definitions
    introduce extraneous technical elements
    which clutter up your thinking
    about the object they define.

     

     

    I will say more about this when I give the various definitions that are in use.

History

Until late in the nineteenth century, functions were usually thought of as defined by formulas (including infinite series). Problems arose in the theory of harmonic analysis which made mathematicians require a more general notion of function. They came up with the concept of function as a set of ordered pairs with the functional property (discussed below), and that understanding revolutionized our understanding of math.

This discussion is an over­simpli­fication of the history of mathe­matics, which many people have written thick books about. A book relevant to these ideas is Plato's Ghost, by Jeremy Gray.

In particular, this definition, along with the use of set theory, enabled abstract math (ahem) to become a common tool for understanding math and proving theorems. It is conceivable that some of you may wish it hadn't. Well, tough.

The more modern definition of function given here (which builds on the older definition) came into use beginning in the 1950's. The strict version became necessary in algebraic topology and is widely used in many fields today.

The concept of function as a formula never disappeared entirely, but was studied mostly by logicians who generalized it to the study of function-as-algorithm. Of course, the study of algorithms is one of the central topics of modern computing science, so the notion of function-as-formula (updated to function-as-algorithm) has achieved a new importance in recent years.

To state both the old abstract definition and the modern one, we need a preliminary idea.

The functional property

A set $P$ of ordered pairs has the functional property if two pairs in $P$ with the same first coordinate have to have the same second coordinate (which means they are the same pair). In other words, if $(x,a)$ and $(x,b)$ are both in $P$, then $a=b$.

How to think about the functional property

The point of the functional property is that for any pair in the set of ordered pairs, the first coordinate determines what the second one is. That's why you can write "$G(x)$'' for any $x $ in the domain of $G$ and not be ambiguous.

Examples

  • The set $\{(1,2), (2,4), (3,2), (5,8)\}$ has the functional property, since no two different pairs have the same first coordinate. Note that there are two different pairs with the same second coordinate. This is irrelevant to the functional property.
  • The set $\{(1,2), (2,4), (3,2), (2,8)\}$ does not have the functional property. There are two different pairs with first coordinate 2.
  • The empty set $\emptyset$ has the function property vacuously.

Graph of a function.

Example: graph of a function defined by a formula

In calculus books, a picture like this one (of part of $y=x^2+2x+5$) is called a graph. Here I use the word "graph" to denote the set of ordered pairs \[\left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in \mathbb{R } \right\}\] which is a mathematical object rather than some ink on a page or pixels on a screen.

The graph of any function studied in beginning calculus has the functional property. For example, the set of ordered pairs above has the functional property because if $x$ is any real number, the formula ${{x}^{2}}+2x+5$ defines a specific real number.

  • if $x = 0$, then ${{x}^{2}}+2x+5=5$, so the pair $(0, 5)$ is an element of the graph of $G$. Each time you plug in $0$ in the formula you get 5.
  • if $x = 1$, then ${{x}^{2}}+2x+5=8$.
  • if $x = -2$, then ${{x}^{2}}+2x+5=5$.

You can measure where the point $\{-2,5\}$ is on the (picture of) the graph and see that it is on the blue curve as it should be. No other pair whose first coordinate is $-2$ is in the graph of $G$, only $(-2, 5)$. That is because when you plug $-2$ into the formula ${{x}^{2}}+2x+5$, you get $5$ and nothing else. Of course, $(0, 5)$ is in the graph, but that does not contradict the functional property. $(0, 5)$ and $(-2, 5)$ have the same second coordinate, but that is OK.

Modern mathematical definition of function

A function $f$ is a mathematical structure consisting of the following objects:

  • A set called the domain of $f$, denoted by $\text{dom} f$.
  • A set called the codomain of $f$, denoted by $\text{cod} f$.
  • A set of ordered pairs called the graph of $ f$, with the following properties:
  • $\text{dom} f$ is the set of all first coordinates of pairs in the graph of $f$.
  • Every second coordinate of a pair in the graph of $f$ is in $\text{cod} f$ (but $\text{cod} f$ may contain other elements).
  • The graph of $f$ has the functional property.

Using arrow notation, this implies that $f:A\to B$.

Remark

The main difference between the specification of function given previously and this definition is that the definition replaces the statement "$f$ has a value at $a$" by introducing a set of ordered pairs (the graph) with the functional property.

  • This set of ordered pairs is extra structure introduced by the definition mainly in order to make the definition a classical sets-with-structure, which makes the graph, which should be a concept derived from the concept of function, into an apparently necessary part of the function.
  • That suggests incorrectly that the graph is more of a primary intuition that other intuitions such as function as relocator, function as transformer, and other points of view discussed in the article Intuitions and metaphors for functions.

Examples

  • Let $F$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ and define $A = \{1, 2, 3, 5\}$ and $B = \{2, 4, 8\}$. Then $F:A\to B$ is a function. In speaking, we would usually say, "$F$ is a function from $A$ to $B$."
  • Let $G$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ (same as above), and define $A = \{1, 2, 3, 5\}$ and $C = \{2, 4, 8, 9, 11, \pi, 3/2\}$. Then $G:A\to C$ is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in $C$, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
  • Let $H$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Then $H:A\to \mathbb{R}$ is a function, since $2$, $4$ and $8$ are all real numbers.
  • Let $D = \{1, 2, 5\}$ and $E = \{1, 2, 3, 4, 5\}$. Then there is no function $D\to A$ and no function $E\to A$ with graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Neither $D$ nor $E$ has exactly the same elements as the first coordinates of the graph.

Identity and inclusion

Suppose we have two sets  A and  B with $A\subseteq B$.

  • The identity function on A is the function ${{\operatorname{id}}_{A}}:A\to A$ defined by ${{\operatorname{id}}_{A}}(x)=x$ for all $x\in A$. (Many authors call it ${{1}_{A}}$).
  • When $A\subseteq B$, the inclusion function from $A$ to $B$ is the function $i:A\to B$ defined by $i(x)=x$ for all $x\in A$. Note that there is a different function for each pair of sets $A$ and $B$ for which $A\subseteq B$. Some authors call it ${{i}_{A,\,B}}$ or $\text{in}{{\text{c}}_{A,\,B}}$.

The identity function and an inclusion function for the same set $A$ have exactly the same graph, namely $\left\{ (a,a)|a\in A \right\}$. More about this below.

Other definitions of function

Original abstract definition of function

Definition

Remarks

Possible confusion

Some confusion can result because of the presence of these two different definitions.

Multivalued function

Some older mathematical papers in com­plex func­tion theory do not tell you that their functions are multi­valued. There was a time when com­plex func­tion theory was such a Big Deal in research mathe­matics that the phrase "func­tion theory" meant complex func­tion theory and all the cogno­scenti knew that their functions were multi­valued.

The phrase multivalued function refers to an object that is like a function $f:S\to T$ except that for $s\in S$, $f(s)$ may denote more than one value.

Examples

  • Multivalued functions arose in considering complex functions. In common practice, the symbol $\sqrt{4}$ denoted $2$, although $-2$ is also a square root of $4$. But in complex function theory, the square root function takes on both the values $2$ and $-2$. This is discussed in detail in Wikipedia.
  • The antiderivative is an example of a multivalued operator. For any constant $C$, $\frac{x^3}{3}+C$ is an antiderivative of $x^2$.

A multivalued function $f:S\to T$ can be modeled as a function with domain $S$ and codomain the set of all subsets of $T$. The two meanings are equivalent in a strong sense (naturally equivalent}). Even so, it seems to me that they represent two differ­ent ways of thinking about multivalued functions. ("The value may be any of these things…" as opposed to "The value is this whole set of things.")

The phrases "multivalued function" and "partial function" upset some picky types who say things like, "But a multi­valued func­tion is not a func­tion!". A step­mother is not a mother, either. See the Hand­book article on radial category.

Partial function

A partial function $f:S\to T$ is just like a function except that its input may be defined on only a subset of $S$. For example, the function $f(x)=\frac{1}{x}$ is a partial function from the real numbers to the real numbers.

This models the behavior of computer programs (algorithms): if you consider a program with one input and one output as a function, it may not be defined on some inputs because for them it runs forever (or gives an error message).

In some texts in computing science and mathematical logic, a function is by convention a partial function, and this fact may not be mentioned explicitly, especially in research papers.

New approaches to functions

All the definitions of function given here produce mathematical structures, using the traditional way to define mathematical objects in terms of sets. Such definitions have disadvantages.

Mathematicians have many ways to think about functions. That a function is a set of ordered pairs with a certain property (functional) and possibly some ancillary ideas (domain, codomain, and others) is not the way we usually think about them$\ldots$Except when we need to reduce the thing we are studying to its absolutely most abstract form to make sure our proofs are correct. That most abstract form is what I have called the rigorous view or the dry bones and it is when that reasoning is needed that the sets-with-structure approach has succeeded.

Our practice of abstraction has led us to new approaches to talking about functions. The most important one currently is category theory. Roughly, a category is a bunch of objects together with some arrows going between them that can be composed head to tail. Functions between sets are examples of this: the sets are the objects and the functions the arrows.

This abstracts the idea of function in a way that brings out common ideas in various branches of math. Research papers in many branches of mathematics now routinely use the language of category theory. Categories now appear in some undergraduate math courses, meaning that Someone needs to write a chapter on category theory for abstractmath.org.

Besides category theory, computing scientists have come up with other abstract ways of dealing with functions, for example type theory. It has not come as far along as category theory, but has shown recent signs of major progress.

Both category theory and type theory define math objects in terms of their effect on and relationship with other math objects. This makes it possible to do abstract math entirely without using sets-with-structure as a means of defining concepts.

 

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Conceptual blending

This post uses MathJax.  If you see formulas in unrendered TeX, try refreshing the screen.

A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another.  Conceptual blending is a major tool used by our brain to understand the world.

The concept of conceptual blend includes special cases, such as representations, images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned.  The Wikipedia article is a good starting place for understanding conceptual blending. 

In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus.  I omit the connections with programs, which I will discuss in a separate post.

A particular function

Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.

FORMULA:

$h(t)$ is defined by the formula $4-(t-2)^2$.

  • The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
  • The formula defines the function, which is a stronger statement than saying it represents the function.
  • The formula is in standard algebraic notation. (See Note 1)
  • To use the formula requires one of these:
    • Understand and use the rules of algebra
    • Use a calculator
    • Use an algebraic programming language. 
  • Other formulas could be used, for example $4t-t^2$.
    • That formula encapsulates a different computation of the value of $h$.

TREE: 

$h(t)$ is also defined by this tree (right).
  • The tree makes explicit the computation needed to evaluate the function.
  • The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
  • Both formula and tree require knowledge of conventions.
  • The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
  • Other formulas correspond to other trees.  In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
  • More about trees in these posts:

GRAPH:

$h(t)$ is represented by its graph (right). (See note 2.)

  • This is the graph as visual image, not the graph as a set of ordered pairs.
  • The blending of graph and formula associates each point on the (blue) graph with the value of the formula at the number on the x-axis directly underneath the point.
  • In contrast to the formula, the graph does not define the function because it is a physical picture that is only approximate.
  • But the formula does represent the function.  (This is "represents" in the sense of cognitive psychology, but not in the mathematical sense.)
  • The blending requires familiarity with the conventions concerning graphs of functions. 
  • It sets into operation the vision machinery of your brain, which is remarkably elaborate and powerful.
    • Your visual machinery allows you to see instantly that the maximum of the curve occurs at about $t=2$. 
  • The blending leaves out many things.
    • For one, the graph does not show the whole function.  (That's another reason why the graph does not define the function.)
    • Nor does it make it obvious that the rest of the graph goes off to negative infinity in both directions, whereas that formula does make that obvious (if you understand algebraic notation).  

GEOMETRIC

The graph of $h(t)$ is the parabola with vertex $(2,4)$, directrix $x=2$, and focus $(2,\frac{3}{4})$. 

  • The blending with the graph makes the parabola identical with the graph.
  • This tells you immediately (if you know enough about parabolas!) that the maximum is at $(2,4)$ (because the directrix is vertical).
  • Knowing where the focus and directrix are enables you to mechanically construct a drawing of the parabola using a pins, string, T-square and pencil.  (In the age of computers, do you care?)

HEIGHT:

$h(t)$ gives the height of a certain projectile going straight up and down over time.

  • The blending of height and graph lets you see instantly (using your visual machinery) how high the projectile goes. 
  • The blending of formula and height allows you to determing the projectile's velocity at any point by taking the derivative of the function.
  • A student may easily be confused into thinking that the path of the projectile is a parabola like the graph shown.  Such a student has misunderstood the blending.

KINETIC:

You may understand $h(t)$ kinetically in various ways.

  • You can visualize moving along the graph from left to right, going, reaching the maximum, then starting down.
    • This calls on your experience of going over a hill. 
    • You are feeling this with the help of mirror neurons.
  • As you imagine traversing the graph, you feel it getting less and less steep until it is briefly level at the maximum, then it gets steeper and steeper going down.
    • This gives you a physical understanding of how the derivative represents the slope.
    • You may have seen teachers swooping with their hand up one side and down the other to illustrate this.
  • You can kinetically blend the movement of the projectile (see height above) with the graph of the function.
    • As it goes up (with $t$ increasing) the projectile starts fast but begins to slow down.
    • Then it is briefly stationery at $t=2$ and then starts to go down.
    • You can associate these feelings with riding in an elevator.
      • Yes, the elevator is not a projectile, so this blending is inaccurate in detail.
    • This gives you a kinetic understanding of how the derivative gives the velocity and the second derivative gives the acceleration.

OBJECT:

The function $h(t)$ is a mathematical object.

  • Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$. 
  • Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
  • The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
  • When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
    • The mathematical object $h(t)$ is a particular set of ordered pairs. 
    • It just sits there.
    • When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
    • I did not say that math objects are inert and timeless, I said you think of them that way.  This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].

DEFINITION

definition of the concept of function provides a way of thinking about the function.

  • One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
  • A concept can have many different definitions.
    • A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties.  But it can be defined as a set with a ternary operation, as well.
    • A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties.  But a partition is an equivalence relation and an equivalence relation is a partition.  You have just picked different primitives to spell out the definition. 
    • If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
      • For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
      • The definition of group doesn't mention that it has linear representations.

SPECIFICATION

A function can be given as a specification, such as this:

If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.

  • This tells you everything you need to know to use the function $h$.
  • It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.

Notes

1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.

2. There are various ways to give a pictorial image of the function.  The usual way to do this is presenting the graph as shown above.  But you can also show its cograph and its endograph, which are other ways of representing a function pictorially.  They  are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:

3. See How to understand conditionals in the abstractmath article on conditionals.

References

  1. Conceptual blending (Wikipedia)
  2. Conceptual metaphors (Wikipedia)
  3. Definitions (abstractmath)
  4. Embodied cognition (Wikipedia)
  5. Handbook of mathematical discourse (see articles on conceptual blendmental representationrepresentation, and metaphor)
  6. Images and Metaphors (article in abstractmath)
  7. Links to G&G posts on representations
  8. Metaphors in Computing Science (previous post)
  9. Mirror neurons (Wikipedia)
  10. Representations and models (article in abstractmath)
  11. Representations II: dry bones (article in abstractmath)
  12. The transition to formal thinking in mathematics, David Tall, 2010
  13. What is the object of the encapsulation of a process? Tall et al., 2000.

 

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The Mathematical Definition of Function

Introduction

This post is a completely rewritten version of the abstractmath article on the definition of function. Like every part of abstractmath, the chapter on functions is designed to get you started thinking about functions. It is no way complete. Wikipedia has much more complete coverage of mathematical functions, but be aware that the coverage is scattered over many articles.

The concept of function in mathematics is as important as any mathematical idea. The mathematician’s concept of function includes the kinds of functions you studied in calculus but is much more abstract and general. If you are new to abstract math you need to know:

  • The precise meaning of the word “function” and other concepts associated with functions. That’s what this section is about.
  • Notation and terminology for functions. (That will be a separate section of abstractmath.org which I will post soon.)
  • The many different kinds of functions there are. (See Examples of Functions in abmath).
  • The many ways mathematicians think about functions. The abmath article Images and Metaphors for Functions is a stub for this.

I will use two running examples throughout this discussion:

  • {F} is the function defined on the set {\left\{1,\,2,3,6 \right\}} as follows: {F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1}. This is a function defined on a finite set by explicitly naming each value.
  • {G} is the real-valued function defined by the formula {G(x)={{x}^{2}}+2x+5}.

Specification of function

We start by giving a specification of “function”. (See the abstractmath article on specification.) After that, we get into the technicalities of the definitions of the general concept of function.

Specification: A function {f} is a mathematical object which determines and is completely determined bythe following data:

  • {f} has a domain, which is a set. The domain may be denoted by {\text{dom }f}.
  • {f} has a codomain, which is also a set and may be denoted by {\text{cod }f}.
  • For each element {a} of the domain of {f}, {f} has a value at {a}, denoted by {f(a)}.
  • The value of {f} at {a} is completely determined by {a} and {f} .
  • The value of {f} at {a} must be an element of the codomain of {f}.

The operation of finding {f(a)} given {f} and {a} is called evaluation.

Examples

  • The definition above of the finite function {F} specifies that the domain is the set {\left\{1,\,2,\,3,\,6 \right\}}. The value of {F} at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that {F(2) = 3}. The codomain of {F} is not specified, but must include the set {\{1,2,3\}}.
  • The definition of {G} above gives the value at each element of the domain by a formula. The value at 3, for example, is {G(3)=3^2+2\cdot3+5=20}. The definition does not specify the domain or the codomain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is {{\mathbb R}}. The codomain must include all real numbers greater than or equal to 4. (Why?)

Comment: The formula above that defines the function G in fact defines a function of complex numbers (even quaternions).

Definition of function

In the nineteenth century, mathematicians realized that it was necessary for some purposes (particularly harmonic analysis) to give a mathematical definition of the concept of function. A stricter version of this definition turned out to be necessary in algebraic topology and other fields, and that is the one I give here.

To state this definition we need a preliminary idea.

The functional property

A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).

Examples

  • The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, since no two different pairs have the same first coordinate. It is true that two of them have the same second coordinate, but that is irrelevant.
  • The set {\{(1,2), (2,4), (3,2), (2,8)\}} does not have the functional property. There are two different pairs with first coordinate 2.
  • The graphs of functions in beginning calculus have the functional property.
  • The empty set {\emptyset} has the functional property .

Example: Graph of a function defined by a formula

The graph of the function {G} given above has the functional property. The graph is the set

\displaystyle \left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in {\mathbb R} \right\}.

If you repeatedly plug in one real number over and over, you get out the same real number every time. Example:

  • if {x = 0}, then {{{x}^{2}}+2x+5=5}.  You get 5 every time you plug in 0.
  • if {x = 1}, then {{{x}^{2}}+2x+5=8}.
  • if {x =-2}, then {{{x}^{2}}+2x+5=5}.

This set has the functional property because if {x} is any real number, the formula {{{x}^{2}}+2x+5} defines a specific real number. (This description of the graph implicitly assumes that {\text{dom } G={\mathbb R}}.)  No other pair whose first coordinate is {-2} is in the graph of {G}, only {(-2, 5)}. That is because when you plug {-2} into the formula {{{x}^{2}}+2x+5}, you get {5} every time. Of course, {(0, 5)} is in the graph, but that does not contradict the functional property. {(0, 5)} and {(-2, 5)} have the same second coordinate, but that is OK.

How to think about the functional property

The point of the functional property is that for any pair in the set of ordered pairs, the first coordinate determines what the second one is. That’s why you can write “{G(x)}” for any {x } in the domain of {G} and not be ambiguous.

Mathematical definition of function

A function{f} is a mathematical structure consisting of the following objects:

  • A set called the domain of {f}, denoted by {\text{dom } f}.
  • A set called the codomain of {f}, denoted by {\text{cod } f}.
  • A set of ordered pairs called the graph of { f}, with the following properties:
    • {\text{dom } f} is the set of all first coordinates of pairs in the graph of {f}.
    • Every second coordinate of a pair in the graph of {f} is in {\text{cod } f} (but {\text{cod } f} may contain other elements).
    • The graph of {f} has the functional property. Using arrow notation, this implies that {f:A\rightarrow B}.

Examples

  • Let {F} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} and define {A = \{1, 2, 3, 5\}} and {B = \{2, 4, 8\}}. Then {F:A\rightarrow B} is a function.
  • Let {G} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} (same as above), and define {A = \{1, 2, 3, 5\}} and {C = \{2, 4, 8, 9, 11, \pi, 3/2\}}. Then {G:A\rightarrow C} is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in {C}, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
  • Let {H} have graph {\{(1,2), (2,4), (3,2), (5,8)\}}. Then {H:A\rightarrow {\mathbb R}} is a function.

According to the definition of function, {F}, {G} and {H} are three different functions.

Identity and inclusion

Suppose we have two sets A and B with {A\subseteq B}.

  • The identity function on A is the function {{{\text{id}}_{A}}:A\rightarrow A} defined by {{{\text{id}}_{A}}(x)=x} for all{x\in A}. (Many authors call it {{{1}_{A}}}).
  • The inclusion function from A to B is the function {i:A\rightarrow B} defined by {i(x)=x} for all {x\in A}. Note that there is a different function for each pair of sets A and B for which {A\subseteq B}. Some authors call it {{{i}_{A,\,B}}} or {\text{in}{{\text{c}}_{A,\,B}}}.

Remark The identity function and an inclusion function for the same set A have exactly the same graph, namely {\left\{ (a,a)|a\in A \right\}}.

Graphs and functions

  • If {f} is a function, the domain of {f} is the set of first coordinates of all the pairs in {f}.
  • If {x\in \text{dom } f}, then {f(x)} is the second coordinate of the only ordered pair in {f} whose first coordinate is {x}.

Examples

The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, so it is the graph of a function. Call the function {H}. Then its domain is {\{1,2,3,5\}} and {H(1) = 2} and {H(2) = 4}. {H(4)} is not defined because there is no ordered pair in H beginning with {4} (hence {4} is not in {\text{dom } H}.)

I showed above that the graph of the function {G}, ordinarily described as “the function {G(x)={{x}^{2}}+2x+5}”, has the functional property. The specification of function requires that we say what the domain is and what the value is at each point. These two facts are determined by the graph.

Other definitions of function

Because of the examples above, many authors define a function as a graph with the functional property. Now, the graph of a function {G} may be denoted by {\Gamma(G)}.  This is an older, less strict definition of function that doesn’t work correctly with the concepts of algebraic topology, category theory, and some other branches of mathematics.

For this less strict definition of function, {G=\Gamma(G)}, which causes a clash of our mental images of “graph” and “function”. In every important way except the less-strict definition, they ARE different!

A definition is a device for making the meaning of math technical terms precise. When a mathematician think of “function” they think of many aspects of functions, such as a map of one shape into another, a graph in the real plane, a computational process, a renaming, and so on. One of the ways of thinking of a function is to think about its graph. That happens to be the best way to define the concept of function.  (It is the less strict definition and it is a necessary concept in the modern definition given here.)

The occurrence of the graph in either definition doesn’t make thinking of a function in terms of its graph the most important way of visualizing  it. I don’t think it is even in the top three.

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Templates in mathematical practice

This post is a first pass at what will eventually be a section of abstractmath.org. It’s time to get back to abstractmath; I have been neglecting it for a couple of years.

What I say here is based mainly on my many years of teaching discrete mathematics at Case Western Reserve University in Cleveland and more recently at Metro State University in Saint Paul.

Beginning abstract math

College students typically get into abstract math at the beginning in such courses as linear algebra, discrete math and abstract algebra. Certain problems that come up in those early courses can be grouped together under the notion of (what I call) applying templates [note 0]. These are not the problems people usually think about concerning beginners in abstract math, of which the following is an incomplete list:

The students’ problems discussed here concern understanding what a template is and how to apply it.

Templates can be formulas, rules of inference, or mini-programs. I’ll talk about three examples here.

The template for quadratic equations

The solution of a real quadratic equation of the form {ax^2+bx+c=0} is given by the formula

\displaystyle  x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

This is a template for finding the roots of the equations. It has subtleties.

For example, the numerator is symmetric in {a} and {c} but the denominator isn’t. So sometimes I try to trick my students (warning them ahead of time that that’s what I’m trying to do) by asking for a formula for the solution of the equation {a+bx+cx^2=0}. The answer is

\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2c}

I start writing it on the board, asking them to tell me what comes next. When we get to the denominator, often someone says “{2a}”.

The template is telling you that the denominator is 2 times the coefficient of the square term. It is not telling you it is “{a}”. Using a template (in the sense I mean here) requires pattern matching, but in this particular example, the quadratic template has a shallow incorrect matching and a deeper correct matching. In detail, the shallow matching says “match the letters” and the deep matching says “match the position of the letters”.

Most of the time the quadratic being matched has particular numbers instead of the same letters that the template has, so the trap I just described seldom occurs. But this makes me want to try a variation of the trick: Find the solution of {3+5x+2x^2=0}. Would some students match the textual position (getting {a=3}) instead of the functional position (getting {a=5})? [Note [0]). If they did they would get the solutions {(-1,-\frac{2}{3})} instead of {(-1,-\frac{3}{2})}.

Substituting in algebraic expressions have other traps, too. What sorts of mistakes would students have solving {3x^2+b^2x-5=0}?

Most students on the verge of abstract math don’t make mistakes with the quadratic formula that I have described. The thing about abstract math is that it uses more sophisticated templates

  • subject to conditions
  • with variations
  • with extra levels of abstraction

The template for proof by induction

This template gives a method of proof of a statement of the form {\forall{n}\mathcal{P}(n)}, where {\mathcal{P}} is a predicate (presumably containing {n} as a variable) and {n} varies over positive integers. The template says:

Goal: Prove {\forall{n}\mathcal{P}(n)}.

Method:

  • Prove {\mathcal{P}(1)}
  • For an arbitrary integer {n>1}, assume {\mathcal{P}(n)} and deduce {\mathcal{P}(n+1)}.

For example, to prove {\forall n (2^n+1\geq n^2)} using the template, you have to prove that {2^2+1\geq  1^1}, and that for any {n>1}, if {2^n+1\geq n^2}, then {2^{n+1}+1\geq  (n+1)^2}. You come up with the need to prove these statements by substituting into the template. This template has several problems that the quadratic formula does not have.

Variables of different types

The variable {n} is of type integer and the variable {\mathcal{P}} is of type predicate [note 0]. Having to deal with several types of variables comes up already in multivariable calculus (vectors vs. numbers, cross product vs. numerical product, etc) and they multiply like rabbits in beginning abstract math classes. Students sometimes write things like “Let {\mathcal{P}=n+1}”. Multiple types is a big problem that math ed people don’t seem to discuss much (correct me if I am wrong).

Free and bound

The variable {n} occurs as a bound variable in the Goal and a free variable in the Method. This happens in this case because the induction step in the Method originates as the requirement to prove {\forall  n(\mathcal{P}(n)\rightarrow\mathcal{P}(n+1))}, but as I have presented it (which seems to be customary) I have translated this into a requirement based on modus ponens. This causes students problems, if they notice it. (“You are assuming what you want to prove!”) Many of them apparently go ahead and produce competent proofs without noticing the dual role of {n}. I say more power to them. I think.

The template has variations

  • You can start the induction at other places.
  • You may have to have two starting points and a double induction hypothesis (for {n-1} and {n}). In fact, you will have to have two starting points, because it seems to be a Fundamental Law of Discrete Math Teaching that you have to talk about the Fibonacci function ad nauseam.
  • Then there is strong induction.

It’s like you can go to the store and buy one template for quadratic equations, but you have to by a package of templates for induction, like highway engineers used to buy packages of plastic French curves to draw highway curves without discontinuous curvature.

The template for row reduction

I am running out of time and won’t go into as much detail on this one. Row reduction is an algorithm. If you write it up as a proper computer program there have to be all sorts of if-thens depending on what you are doing it for. For example if want solutions to the simultaneous equations

2x+4y+z = 1
x+2y = 0
x+2y+4z = 5

you must row reduce the matrix

2 4 1 1
1 2 0 0
1 2 4 5

(I haven’t yet figured out how to wrap this in parentheses) which gives you

1 2 0 0
0 0 1 0
0 0 0 1

This introduces another problem with templates: They come with conditions. In this case the condition is “a row of three 0s followed by a nonzero number means the equations have no solutions”. (There is another condition when there is a row of all 0’s.)

It is very easy for the new student to get the calculation right but to never sit back and see what they have — which conditions apply or whatever.

When you do math you have to repeatedly lean in and focus on the details and then lean back and see the Big Picture. This is something that has to be learned.

What to do, what to do

I have recently experimented with being explicit about templates, in particular going through examples of the use of a template after explicitly stating the template. It is too early to say how successful this is. But I want to point out that even though it might not help to be explicit with students about templates, the analysis in this post of a phenomenon that occurs in beginning abstract math courses

  • may still be accurate (or not), and
  • may help teachers teach such things if they are aware of the phenomenon, even if the students are not.

Notes

  1. Many years ago, I heard someone use the word “template” in the way I am using it now, but I don’t recollect who it was. Applied mathematicians sometimes use it with a meaning similar to mine to refer to soft algorithms–recipes for computation that are not formal algorithms but close enough to be easily translated into a sufficiently high level computer language.
  2. In the formula {ax^2+bx+c}, the “{a}” has the first textual position but the functional position as the coefficient of the quadratic term. This name “functional position” has nothing to do with functions. Can someone suggest a different name that won’t confuse people?
  3. I am using “variable” the way logicians do. Mathematicians would not normally refer to “{\mathcal{P}}” as a variable.
  4. I didn’t say anything about how templates can involve extra layers of abstract.  That will have to wait.
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