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This Glossary is an alphabetized list of words and phrases used in Mathematical English. It includes several kinds of entries:
Most technical words in math are not included. The best sources for technical words and phrases in math are:
The word "a" or "an" is the indefinite article, one of two articles in English.
The articles in English are the indefinite article "a" (with variant "an") and the definite article "the". If your native language is not English you may have problems understanding mathematical discourse because the articles are used in mathematical English in some fairly subtle ways.
In math writing, the indefinite article may be used with the name of a type of math object to produce an indefinite description – a description that indicates an arbitrary or unspecified object of that type. (If you use "the", you get a definite description that refers to an instance of the object that has been previously referred to.)
When an indefinite description occurs
in the subject of a clause in math English,
it usually must be read as referring to
every object that fits that description.
"Show that an integer that is divisible by four is divisible by two."
So in a sentence like this the indefinite article has the force of a universal quantifier. Unfortunately, this is also true of the definite article in some circumstances! More examples are given in the entry on universal quantifier.
In ordinary English sentences, such as a "A wolf takes a mate for life." the meaning is that the assertion is true for a typical individual (typical wolf in this case). In math, however, the assertion is required to be true without exception.
See algebra.
Notation may be called abuse of notation if it involves
Click on those entries for examples.
The word "abuse" makes it sound worse than it really is. Without judicious use of this technique much mathematical writing would be unreadable.
This word has many different meanings in the school system of the USA, and college math majors in particular may be confused by the differences.
High school and college math are changing rapidly,
so these descriptions are not dependable.
The word algorithm is used in three confusingly similar ways:
Mathematicians typically use the word "algorithm" for a step by step process for calculating something, as for example the procedure expressed roughly by the description (NM) below. People who use the word "algorithm" in this way may refer to a program implementing it as the code for the algorithm.
A program may itself be called "an algorithm". In my experience, this is not common usage.
In computer science or logic texts, the word "algorithm" may be given a mathematical definition (for example as a Turing machine (MW, Wik)), turning an algorithm into a mathematical object.
You can write a program in Python and another one in C to take a list $a:=(a_1,\,a_2,\,a_3,\,a_4\,\ldots)$ with at least three entries and swap the second and third entries. There is a sense in which the two programs, although different as programs, implement the "same" algorithm (process): Change $(a_1,\,a_2,\,a_3,\,a_4\,\ldots)$ to $(a_1,\,a_3,\,a_2,\,a_4\,\ldots)$.
Newton’s Method gives rise to an algorithm in the sense of a process.
(NM) Start with a guess $x$ and let $x:=x-f(x)/f'(x)$ (see colon-equals) repeatedly until either
This algorithm is not hard to implement in C, Pascal, Python or many other programming languages. But in all these cases you have to get the syntax exactly right and take care of a lot of details such as assigning labels to certain lines of the program. Note that (NM) is not any of those programs: it is the process carried out by those programs.
Algorithms are discussed in Functions: images and metaphors and in detail in Wikipedia.
For many choices of starting point, Newton's algorithm will run forever and not give an answer. (In practice it would probably result in some kind of overflow and stop with an error message.) For many authors, "algorithm" means it always stops in a finite number of steps. For them, Newton's process is not an algorithm. There is a problem with their attitude: Determining whether a process is an algorithm is an undecidable problem.
The symmetry of the square illustrated by the figure below can be described in two different ways.
"Alibi" and "alias" are not mathematical properties of
transformations;
they are ways to think
about them.
See universally true and universal conditional.
The statement "$x^2+1$ is always positive" (for a real variable $x$) is equivalent to the universally true statement, "For every real number $x$, $x^2+1$ is positive" (strictly greater than $0$.)
This use of "always" is noticeably less formal than other ways of expression universal statements. The mental image behind it is that you can vary $x$ all over the place as long as you want and the expression stays greater than $0$.s
The word ambient is used to refer to a mathematical object such as a space that contains a given mathematical object. It is also commonly used to refer to an operation on the ambient space.
"Let $A$ and $B$ be subspaces of a space $S$ and suppose $\phi$ is an ambient homeomorphism taking $A$ to $B$." The point of this sentence is that $A$ and $B$ are not merely homeomorphic, but they are homeomorphic via the automorphism $\phi$ of the space $S$.
Used to emphasize that there is no restriction on the mathematical structure referred to by the noun phrase that follows. In most cases the word adds no additional mathematical meaning to the statement. You may usually use "any" in this situation instead of "arbitrary".
"The equation ${{x}^{r}}{{x}^{s}}={{x}^{r+s}}$ holds in an arbitrary group (or "in any group"), but the equation ${{x}^{r}}{{y}^{r}}={{(xy)}^{r}}$ requires commutativity."
In a phrase such as "Let S be an arbitrary set" the word arbitrary typically signals an expectation of an upcoming proof by universal generalization.
People new to abstract math may have a systematic tendency to underestimate how arbitrary a math object can be. For example, the set $\{3,\,\,\pi ,\,\,\mathbb{R},\,\text{ the cosine function}\}$ is a perfectly good set. It is arbitrary, and, I admit, weird, but it is a set. Other examples:
More arbitrary sets.
The function in the Examples of Functions section called a finite function is an arbitrary function.
This word has three common meanings in mathematical discourse.
This word can cause cognitive dissonance. In English,"argument" can mean either:
The meaning of disagreement is the common one and it may carry a connotation of unpleasantness. The three meanings in math that are given above have no connotation of unpleasantness.
An assertion in a mathematical text is a statement in math English (more here) or the symbolic language (see symbolic assertion). The author’s intention in writing an assertion may be difficult to determine (see intent).
An assumption is an assertion that is taken as true in a given block of text that is its scope. "Taken as true" means that any proof in the scope of the assumption may use the assumption to justify a claim without further argument.
For real numbers $x$ and $y$ the phrase"$x$ is at least $y$" means $x\ge y$. The phrase " $x$ is at most $y$" means $x\le y$.
A line drawn over a single symbol is pronounced "bar". For example, $\overline{x}$ is pronounced "$x$ bar". It often refers to the closure (in some sense) of a subset of a set.
"Let $X$ be a topological space with subset $Y$, and let $\overline{Y}$be the closure of $Y$ in $X$."
Other names for this symbol are macron and vinculum.
The verb "to be" with its inflected forms "is" and "are" has many uses in the English language (see list in Wiktionary). Here I mention a few common usages in mathematical texts.
For example,"Every integer is a rational number."
"Every integer is rational." This means the same thing as "Every integer is a rational number", but is expressed using different grammar. See property.
In defining a property, the word "is" may connect the definiendum to the name of the property, as in:
"A group is Abelian if $xy=yx$ for all elements $x$ and $y$."
Note that this is not an assertion that some group is Abelian, as in the previous entry; instead, it is saying what it means to be Abelian. It is not always easy to know whether the author means such a statement as an assertion or as a definition. In this example the fact that the word "Abelian" is in boldface communicates that it is a definition. See definitions for more about this.
In statements such as:
"A semigroup is a set with an associative multiplication defined on it."
the word "is" connects a definiendum with the condition defining it. Note again that it may not be clear whether this is an assertion or a definition. See definitions for other examples.
The word "is" in the statement
"An idempotent function has the property that its image is its set of fixed points."
asserts that two mathematical descriptions ("its image" and "its set of fixed points") denote the same mathematical object. This is the same as the meaning of " =".
For example, "There is a number $x$ which is bigger than $2$ ". See existential quantifier for other ways this can be worded.
This word has several related usages.
In math texts, brackets can mean any of the delimiters parentheses, square brackets, braces, and angle brackets (see delimiters). Some American dictionaries and some mathematicians restrict the meaning to square brackets or angle brackets
The word "bracket" is used in various mathematical specialties as the name of an operator (for example, Lie bracket, Toda bracket, Poisson bracket) on an algebra. The operation called bracket may use square brackets, braces or angle brackets to denote the operation. The Lie bracket of $v$ and $w$ is always (as far as I know) denoted by $[v,w]$. On the other hand, notation for the Poisson and Toda brackets varies, with different authors using different symbols.
The word "bracket" may be used to denote the value of the expression inside a pair of brackets (in the sense of delimiters). For example,
"If the expression \[\left(x^2-2x+1\right)+\left(e^{2x}-5\right)^3\] is zero, then the two brackets are opposite in sign." This usage may be obsolescent.
As a conjunction, but typically means the same as "and", with an indication that what follows is surprising or in contrast to what precedes it. This is a standard usage in English, not peculiar to mathematical English.
" 5 is odd, but 6 is even."
Mathematical authors may begin a sentence with "but" to indicate that the subject under discussion has a relevant property that will now be mentioned. For example, it may be relevant because it leads to the next step in the reasoning. The property may be one that is easy to deduce or one that has already been derived or assumed.
This usage may carry with it no thought of contrast or surprise. Of course, in this usage "but" still means "and" as far as the logic goes; it is the connotations that are different.
"We have now shown that $m = pq$, where $p$ and $q$ are primes. But that implies that $m$ is composite."
(In a situation where we already know that $x=7$):
"... We have now proved that ${{x}^{2}}+{{y}^{2}}=100$. But $x$ is 7, so $y=\sqrt{51}$.
To calculate may mean to perform symbol manipulation on an expression, usually with the intent to arrive at another, perhaps more satisfactory, expression. "Compute" is used in much the same way. Neither word necessarily indicates that the procedure was done on a computer or that a known algorithm was used.
Call may be used to form a definition.
The Roman alphabet, the Greek alphabet, and the Cyrillic alphabet have two forms of letters, "capital" or uppercase, A, B, C, etc, and lowercase, a, b, c, etc.
Case distinction always matters in mathematics. It is common for example to use a capital letter to name a mathematical structure and the same letter in lowercase to name an element of the structure. This means that when you take notes at a lecture in abstract math, you must write "$S$" when the lecturer writes "$S$" and "$s$" when the lecturer writes "$s$"!
Other variations in font and style may also be significant. See fraktur, boldface and blackboard bold.
The word "cases" in this entry has no connection with the word "case" described in the preceding entry.
A concept is defined by cases if it is dependent on a parameter and the defining expression is different for different values of the parameter. This is also called a disjunctive definition or split definition.
Disjunctive definitions may seem unnatural at first. This may be because real life definitions are rarely disjunctive. One exception is the concept of strike in baseball (swing and miss or didn’t swing when the ball was in the zone or one of several other possibilities).
The word category is used with two unrelated meanings in math (Baire and Eilenberg-Mac Lane). It is used with still other meanings by linguists and cognitive scientists.
A set is closed under an operation if the result of applying the operation to elements of the set is again an element of the set.
Much more about this idea is under closure in Wikipedia.
An $r$-combination of a set $S$ is an $r$-element subset of $S$."Combination" is the word used in combinatorics. Everywhere else in mathematics, a subset is called a subset.
A comma between symbolic assertions may denote and.
The set $\{m|m={{n}^{2}},\,\,\,n\in \mathbb{Z}\}$ denotes the set of squares of integers. The defining condition is: "$m={{n}^{2}}$ and $n$ is an integer".
The comma may also be used to indicate a distributive constraint.
"Let $x,\,\,y\ne 0$." This means "Let $x\ne 0$ and $y\ne 0$", which means the same as "Let each of $x$ and $y$ be nonzero".
See calculate.
If $A$ and $B$ are sets, the assertion "$A$ contains $B$" can mean one of two things: $B\subseteq A$ or $B\in A$. The sentence "$B$ is contained in $A$" can mean either of these things as well.
To apply a process coordinatewise to a list or vector is to apply the process to each coordinate individually and independently.
Define means to give a definition.
Some students find the third form – define as a command – to be confusing: "What am I supposed to do when it tells me to define something?" Answer: You don’t do anything, you simply understand that from now on "even" means "divisible by $2$".
The word degenerate is used in some disciplines to describe a particular math structure with one or both of the following properties:
a) Some parts of the structure that are normally distinct are the same in this particular example.
b) Some parameter is 0. (Trivial is also sometimes used for this meaning.)
More than half of the examples I turned up on JSTOR when doing research for the Handbook were references to degenerate critical points.
To say that an expression $E$ denotes a specific object $B$ means that $E$ refers to $B$; in an assertion containing a description of $B$, the description can be replaced by $E$ and the truth value of the assertion remains the same. $B$ is sometimes called the denotation of $E$.
A description (in mathematical English and in the symbolic language) is a noun or noun phrase that refers to a particular object (a definite description) or an unspecified object of a certain type (indefinite description).
Two sets $S$ and $T$ are disjoint if their intersection is empty, in other words if they have no elements in common. A family of sets is pairwise disjoint if any two different sets in the family are disjoint.
"Disjoint" requires two sets. Don’t say things such as: "Each set in a partition is disjoint". You could say "each set in a partition is disjoint from each of the other sets."
When several new identifiers are introduced at once, the word distinct is used to require that no two different ones can have the same value.
"Let $m$ and $n$ be distinct integers." This means that in the text that follows, one can assume that $m\neq n$.
Students may not understand that without a word such as "distinct", the variables may indeed have the same value. Thus "Let $m$ and $n$ be integers" allows $m = n$.
It is reported that E. H. Moore was sufficiently bothered by this phenomenon to say, "Let $m$ be an integer and let $n$ be an integer."
A domain may be any of these:
I recall as a graduate student being puzzled by the first two meanings given above, with the result that I spent a (mercifully short) time trying to prove that the domain of a continuous function had to be a connected open set.
The expression "e.g." means "for example". The expression "i.e." means "that is". These abbreviations confuse many readers and writers, including writers in the research literature. They have are significantly different meanings.
If you desperately want to be understood, |
The word each may generally be used in the sasme way as all, every and any to form a universal quantifier. However it is not common to see an assertion worded as "Each multiple of $4$ is even". "Each" is more commonly used for a distributive plural.
Used to associate the structure attached to a set to make up a mathematical structure. Also endowed.
A semigroup is a set equipped with [endowed with] an associative binary operation.
A factor is a constituent of a product.
The word "product" is used for many other math objects besides integers and polynomials, and there are conventions about the use of "factor" in those cases as well. I have not done any lexicographical research about these other situations, but I know enough examples to assert that the situation is quite complicated. For example, a factor of a group $G$ does not have to occur in a product of groups that is isomorphic to $G$.
A fallacy is an error in reasoning. Two fallacies with standard names that are commonly committed by students are affirming the consequent and denying the antecedent. See also argument by analogy.
Terminology The meaning of fallacy given here is that used in abmath in general. It is widely used with a looser meaning and often connotes deliberate deception, which is not intended here.
A family of sets sometimes means an indexed set (tuple) of sets (so differently indexed members may be the same) and sometimes merely a set of sets.
The two meanings are unrelated, and you often have to tell from the context which one is meant.
The word also refers to specialty, as in "My field is topological semigroups". This meaning occurs in all academic subjects. "Field" is also discussed in the chapter on Names.
Used in problems in much the same way as give.
The statement that an assertion $Q$ follows from an assertion $P$ means that $P$ implies $Q$ (if $P$, then $Q$).
"The integer $n$ is divisible by 16. It follows that it is even."
The word "follow" is also used to indicate that some statements after the current one are to be grouped with the current one, or (as in "the following are equivalent)" are to be grouped with each other.
"A set $G$ with a binary operation is a group if it satisfies the following axioms ... " This statement indicates that the axioms that follow are part of the definition currently in progress.
The phrase "the following are equivalent" (or "TFAE") is used to assert the equivalence of the assertions that follow (presented in a list).
The following are equivalent:
The equation $A=\pi {{r}^{2}}$ is a formula in both senses, but $\pi {{r}^{2}}$("the formula for the area of a circle") is a formula only in the first sense above. In mathematical logic, $\pi {{r}^{2}}$ would be called a term.
In the sentence, "When it is the subject of a clause, an indefinite description has the force of universal quantification", the pronoun "it" refers to the phrase "indefinite description", which occurs later in the sentence. This is called a forward reference (or "cataphoric reference"). Not only non-native English-speakers but many native English speakers do not automatically understand forward references. That is why in the entry for indefinite article, I reworded that sentence.
Another example: The question "Explain how to determine from the last digit of its octal notation whether an integer is even" baffled some of my discrete math students.
Forward reference is a type of coreference (see also "coreference" in the Handbook).
Give is used in several ways in math English. often with the same sense it would be used in any academic text ("we give a proof... ","we give a construction ... ").
To "give an object" means to describe it sufficiently that it is uniquely determined. A phrase of the form "give an $X$ such that $P$" means describe a object of type $X$ that satisfies predicate $P$. The word find can be used in this meaning as well.
"Problem: Give (or find) a function of $x$ that is positive at $x$ = 0." Correct answers to this problem could be "the cosine function on the reals" or "the functions $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)= 2x + 1$."
$ $"Given sets $S$ and $T$, the intersection $S\cap T$ is the set of all objects that are elements of both $S$ and $T$."
This means "If $S$ and $T$ are sets, the intersection $S\cap T$ is the set of all objects that are elements of both $S$ and $T$" or "Let $S$ and $T$ be sets. Then the intersection $S\cap T$ is$\ldots$"
To glue two math objects together is to identify them. You make a Möbius strip by reversing the right edge of a rectangle and glueing it to the left edge, as described here. This means you identify (in the mathematical sense) the reversed right edge with the left edge. This is a metaphor that is also a literal description of what you do with a real rectangle made out of paper, too!
The word graph has two unrelated meanings in undergraduate mathematics:
In both cases, the word graph is used both for the math object itself and also for a drawing of (often only part of) the math object.
An assertion $P$ containing a variable $x$ holds for $a$ if $P$ becomes true when $a$ is substituted for $x$.
These various meanings are quite commonly used without explanation.
To identify a math object $A$ with another object $B$ is to regard them as the same object. In particular each element or point in $A$ is regarded as the same as a particular element or point in $B$. In most cases, the author has in mind a particular way of matching the points in $A$ and the points in $B$. This "particular way" may be presented informally or as a formal mathematical construction.
This is an informal or suggestive definition of "identify", not a math definition, because I haven't said what "element or point in" means.
These are two different ways of thinking about the plane -- as a set of points represented by ordered pairs, or as a set of ordered pairs. This is an example of conceptual blending.
The Möbius strip may be constructed by identifying or gluing the edge \[\left\{ (x,0)|\,0\le x\le 1 \right\}\] of the unit square with the edge \[\left\{ (x,1)|\,0\le y\le 1 \right\}\] in such a way that $(x, 0)$ is identified with $(1 – x, 1)$. This can be thought of as saying that the coordinates $(x, 0)$ and $(1 – x, 1)$ refer to the same point. This is an example of identifying objects by a formal mathematical construction.
This word has three common meanings.
An identity in this sense is an equation that holds between two expressions for any valid values of the variables in the expressions. Thus, for real numbers (in fact for complex numbers), the equation \[{{(x+1)}^{2}}={{x}^{2}}+2x+1\] is an identity. An identity in this sense may also be called a law.
Sometimes in the case of an identity the symbol "$\equiv$" is used instead of the equals sign.
If $x\Delta e=e\Delta x=x$ for all elements $x$ in an algebraic structure with binary operation $\Delta$, then $e$ is an identity or identity element for the structure. Such an element is also called a unit element or unity.
A ring with identity may mean a ring that has an identity element, but it can also mean a ring subject to an algebraic identity.
For a given set $S$, the function from $S$ to $S$ that takes every element of $S$ to itself is called the identity function. (More about that here.) This is an example of a polymorphic definition.
The identity function on $S$ is the identity element of the algebraic structure consisting of all functions from a set $S$ to itself with composition as operation.
The ways if is used in stating conditional assertions is discussed here. See also let.
It is a convention that, in a definition, the word if used to introduce the word being defined means "if and only if".
Some authors regularly use if and only if or "iff" in a definition, so that definition would be worded:
(b) "An integer is even if and only if it is divisible by $2$."
But note: The following statement is not a definition, but an equivalence:
(c) "An integer is divisible by $2$ if and only if it is divisible by $–2$."
Even though "if and only if" is used in both (b) and (c), they don’t have the same status. In statement (b) the left side and the right side are not symmetric. The point of statement (b) is to give a meaning to the left side. Statement (c) is symmetric; you could just as correctly say, "An integer is divisible by $–2$ if and only if it is divisible by $2$ ."
If can be used in the precondition of a definition to introduce some requirements on an object mentioned in the definition. For example, definition (a) above could have been worded this way:
"If $n$ is an integer, then it is even if it is divisible by $2$ ."
If $P$ and $Q$ are two assertions, the statement "$P$ if and only if $Q$" is an assertion that $P$ and $Q$ are equivalent. That statement may be abbreviated "$P$ iff $Q$".
In is used in mathematical discourse in many of its normal English meanings, as well as in some meanings that are peculiar to mathematics.
Let $f:\mathbb{R}\to \mathbb{R}$ be a function.
These words "increasing", "decreasing" and so on can be used for functions to and from other ordered sets, as well.
These words are also used for sequences, since a sequence is a function on its index set. For example, a sequence ${{a}_{1}},\,\,{{a}_{2}},\,\,{{a}_{3}},\,\ldots.$ is increasing if for all $i$ and $j$ in its index set, if $i\lt j$ then ${{a}_{i}}\lt{{a}_{j}}$.
An inequality is an assertion of the form $s\,\alpha \,t$, where $s$ and $t$ are terms and $\alpha$ is one of the relations $\lt$, $\gt$, $\leq $ or $\geq $.
The assertions \[(4+3)^2\gt 8\times 6\] and \[\sin \frac{3\pi}{2}\leq0\] are inequalities.
If you are bothered by the second assertion, see unnecessarily weak assertions.
The word "inequality" is not usually used
to mean an assertion of the form "$s\neq
t$".
The phrase "$A$ inhabits $B$" can mean any of several things:
Lives in is used similarly in conversation, at least for "element of", but I have found only a few citations in print.
Functions can have a property called injective, and so can modules. These two uses are unrelated.
If $f(x)=x^2+1$, then $f(2)=5$.
This word has three different meanings.
Integral is used as an adjective to require that the noun phrase it modifies denote an integer. For example, $8$ is an integral power of $2$ but $9$ is not. This is also used for the more general notion of algebraic integer (MW, Wi).
An integral of a function is an antiderivative of the function. It may be called the indefinite integral to distinguish it from the definite integral mentioned below. The indefinite integral of $f$ may be denoted by $\int{f(x)\,dx}$ which determines the integral only up to a constant.
You may refer to a specific antiderivative using the form $\int_{c}^{t}{f(x)\,dx}$.
The word integral is also used to denote a solution of a more general differential equation.
Integral is also used to denote a definite integral: this operator takes an integrable function and an interval (or more general space) on which the function is defined and produces a number. The definite integral of $f$ on an interval $ [a, b] $ is denoted $\int_{a}^{b}{f(x)\,\,dx}$ or $\int_{a}^{b}{f}$.
Mathematicians may use the word intuition in referring to some image or metaphor they have about a type of object.
See also the discussion of intuition in Diagnostic examples.
See be.
One use of the word just in mathematical discourse is to indicate that what precedes satisfies the statement that occurs after the word "just".
(Assuming $r$ and $s$ are known to be integers greater than $1$).
"... Then $m = rs$. But that just means that $m$ is composite".)
In such sentences, "just" may commonly be omitted without changing the meaning, but in many cases it probably shouldn’t be omitted because it is an important indicator of the logical flow.
Before I did the research (finding citations of usage) for the article on "just" in the Handbook, my own perception of this usage was that the word "just" meant that what followed was equivalent to what preceded. That’s a small example of the vital necessity of checking citations before making pronouncements about math English!
This phrase means that what follows is logically equivalent to what precedes. A search of Jstor reveals that in math research articles it is used mostly but not entirely by logicians.
"An integer is even just in case it is divisible by $2$ "
In ordinary English (used more in Britain than in the USA, I think), "just in case" means "to guard against the possibility of".
"I will take an umbrella just in case it rains" means something like "I will take an umbrella in order to guard against the (undesired) possibility that it might rain."
This can cause cognitive dissonance: The statement "An integer is even just in case it is divisible by $2$ " certainly does not mean "An integer is even in order to guard against the possibility that it might be divisible by $2$ ."
A theorem. Usually when an author calls a theorem a lemma, the connotation is that the lemma is not of interest for itself, but is useful in proving other theorems. However, some lemmas (König's Lemma (MW, Wi), Schanuel’s Lemma (PlanetMath), Zorn's Lemma (MW, Wi)) have become quite famous and are explicitly taught in some courses because of their usefulness.
In old books sometimes the plural of lemma is lemmata.
"Let" is used in several different ways in mathematical English.
The most common use of "let" is to introduce a new symbol or name. This makes it a kind of definition. The scope is usually restricted to the current section of text. In contrast, the scope of a formal definition explicitly using the word "definition" is generally the whole discourse.
There is no logical distinction between this use of "let" and a formal definition. The difference apparently concerns whether the newly introduced expression is for temporary use or meant to hold throughout the text, and perhaps whether it is regarded as important or not.
Consider the theorem
"An integer divisible by $4$ is divisible by $2$." A proof could begin this way: "Let $n$ be an integer divisible by $4$." This introduces a new variable symbol $n$ and constrains it to be divisible by $4$.
Suppose the theorem of the preceding example had been stated this way:
"Let $n$ be an integer. If $n$ is divisible by $4$ then it is divisible by $2$." Then the proof could begin, "Let $n$ be divisible by $4$." In this sentence, $n$ is introduced in the theorem and is further constrained in the proof.
These two examples illustrate that whether a new symbol is introduced or a previous symbol is given a new interpretation is a minor matter of wording; the underlying logical structure of the argument is the same.
"Let $n \gt 0$.... Now let $n\le 0$" If, assume and suppose seem to be more common that "let" in this use.
To pick an unrestricted object from a collection with the purpose of proving an assertion about all elements in the collection using universal generalization. Often used with arbitrary. If, assume and suppose can be used here.
If you are writing a proof of a theorem that claims something is true about all prime numbers, your proof might start out: "Let $n$ be a prime number". During the proof you can use the fact that $n$ is a prime number, but you may not make other special assumptions about it.
To provide a word or symbol for an arbitrary object from a nonempty collection of objects. Equivalently, to choose a witness to an existential assertion that is known to be true. If, assume and suppose can be used here.
In proving a theorem about a differentiable function that is increasing on some interval and decreasing on some other interval, you might write:
"Let $a$ and $b$ be real numbers for which $f'(a)\gt0$ and $f'(b)\lt0$."
These numbers exist by hypothesis.
In the context that $G$ is known to be a noncommutative group:
"Let $x$ and $y$ be elements of $G$ for which $xy\neq yx$.
The following is a more explicit version of the same assertion:
"Let the noncommutative group $G$ be given. Since $G$ is noncommutative, the collection $\left\{ (x,y)\in G\times G|\,xy\ne yx \right\}$ is nonempty, so we may choose a member ($x$,$y$) of this set..."
In proving a function $F:S\to T$ is injective, you might begin with "Let $x,\,\,x'\in S$ be elements for which $F(x)=F(x')$". These elements must exist if $F$ is non-injective: in other words, this begins a proof by contrapositive. The existence statement for which the elements $x,\,\,x'\in S$ are witnesses is implied by the assumption that $F$ is not injective.
Let can be used in the defining phrase of a definition.
"Let an integer be even if it is divisible by 2."
This usage strikes me as unidiomatic. It sounds like a translation of a French ("Soit...") or German ("Sei...") subjunctive. If, assume and suppose cannot be used here.
Assume, define, if, and suppose can be used instead of "let”. The syntax varies depending on which word is used, and some of these words cannot be used in certain situations. There are other subtle differences as well.
You can say,
All these sentences have the same mathematical content.
The periods after "integer" in the first three sentences may be replaced by a semicolon, but not by a comma. The comma in the fourth sentence cannot be replaced by a period; "If $S$ is an odd integer" is not a complete sentence.
The word "affine" is generalized to vector spaces: see MathWorld.
For real numbers $a$ and $x$ with $a$ positive and not equal to 1, the expression ${{\log }_{a}}x$ denotes the number $y$ for which ${{a}^{y}}=x$. The number $a$ is called the base.
You will quite often see the expression "log $x$" with the base omitted (so that the expression has a suppressed parameter.) This expression can mean three different things, depending on the specialty of the text.
Authors
commonly don’t tell you |
To maximize a function is to find values of its argument for which the function has a maximum. Minimize is used similarly.
The number $t=5$ maximizes the function $h(t)=25-{{(t-5)}^{2}}$ (discussed here.)
The metaphor behind this usage seems to be: vary the input until you find the largest value. This is the way most functions are maximized or minimized, using more or less sophisticated methods of varying the input – the bisection method and Newton’s Method are examples of this.
"Mean" may be used in forming a definition.
"To say that an integer is even means that it is divisible by 2."
To say that an assertion $P$ means an assertion $Q$ may signify that implies $Q$.
"We have proved that $4$ divides $n$. This means in particular that n is even."
Math authors do not always make it explicit whether they are using "mean" to give a definition or to describe an implication.
Mean is also a technical term, referring to the arithmetic average.
The word minus can refer to both the binary operation on numbers, as in the expression $a-b$, and the unary operation of taking the negative: negating $b$ gives $-b$. In current usage in American high schools, $a – b$ would be pronounced "a minus b", but $–b$ would be pronounced "negative $b$". The older usage for $-b$ was "minus $b$" and many old fuddy duddy college teachers like me forget and call it "minus $b$" sometimes.
Don’t assume a minus sign before an expression makes it negative. The expression "$–t$" denotes a positive number if $t$ happens to be $–42$. Don’t be misled by the fact that we call it "negative $t$". See also subtraction.
The phrase "$x$ is the same as $y$ modulo $E$" means that $x$ and $y$ are elements of some set, $E$ is an equivalence relation on the set, and $x\, E\, y$. There are variants to this usage:
The word arose from a special equivalence relation in number theory, which has two conflicting special notations in pure math and computing science that I have seen cause confusion among students.
For integers $a$, $b$ and $n$, the expression $a\equiv b\bmod n$ means that $a – b$ is divisible by $n$. (This relation is an equivalence relation. For example, $23\equiv 11\bmod 3$ and $-4\equiv 16\bmod 5$. In this usage, the symbol "mod" occurs as part of a three-place assertion.
In this usage in number theory, the symbol "$\equiv$" does not mean "identical to".
In most computer languages and in computing science texts, the expression "$a\bmod n$" means the least nonnegative remainder obtained when $a$ is divided by $n$. For example, $23 \bmod 3 = 2$ and $-4 \bmod 5 = 1$. (See the Wikipedia article on modulo for details.) In this usage, mod is a binary operation on the integers.
People in math-related fields use "mod" in phrases such as "The administration kept my salary the same modulo [or mod] inflation". Presumably the equivalence relation here is something like: "One dollar in 2002 is equivalent to $1.02 in 2003."
In particular, a statement such as "The alternating group on three letters is the same as the cyclic group of order $3$ up to isomorphism" (less often, "modulo isomorphism") refers to the equivalence relation of two structures being isomorphic.
The word multiplication is the name for many binary operations.
Multiplication of numbers is commutative: For any numbers $a$ and $b$, $ab=ba$. This is not necessarily true in groups. For example, let $a$ be the permutation of the set $\{1,2,3\}$ that switches $1$ and $2$ and let $b$ be the permutation of the set $\{1,2,3\}$ that switches $2$ and $3$. Then the product $ab$ (do $a$ then $b$) takes $1$ to $3$, $3$ to $2$ and $2$ to $1$, whereas the product $ba$ takes $1$ to $2$, $2$ to $3$ and $3$ to $1$.
Some group theory texts write "$ab$" to mean "do $b$ then $a$". Some of them are even nice enough to tell you which way they are doing it.
Countless legions of group theory students have quite unconsciously tried to prove some theorem by writing (for example) $aba=a^2b$ and similar blunders.
The definition of "group" does not require that multiplication be commutative, and:
If a math definition defines a word
that is familiar to you in some other context,
you can't use properties it has in that context in a proof.
You often see "must be" used in math English when "is" would give the same meaning. It is used with verbs other than "be" in the same way. This may be to emphasize that the fact being asserted can be proved from facts known in the context of the discussion.
Other uses of "must" in mathematical discourse are generally examples of the way the words is used in ordinary discourse.
Used to indicate that what follows is one of the following:
$Q$ is necessary for $P$ if $P$ implies $Q$, in other words if the conditional assertion "If $P$, then $Q$" is true. Examples are given under conditional assertion.
The motivation for the word "necessary" is that the assertion "$P$ implies $Q$" is logically equivalent to "not $Q$ implies not $P$" (see contrapositive), so that for $P$ to be true it is necessary in the usual sense of the word for $Q$ to be true.
A statement like "$A$ is never $P$" is used when $A$ depends on a variable (parameter) $x$ and $P$ is a property; it means "For every value of $x$, $A$ does not have property $P$".
See negation.
A real number $r$ is nonnegative if it is not negative, in other words if $r\ge 0$.
See trivial.
Notation is a system of signs and symbols used as a representation of something. The word "notation" may be used to refer to a whole system or to a way of representing a particular object.
Mathematicians may say, "Let's establish some notation", meaning they will introduce a specific combination of certain symbols to refer to a particular mathematical object. This is a type of definition on the fly, so to speak. See also fix and let.
The word "now" is used in at least three different ways in mathematical writing. They are not always easy to distinguish, but only the first usage listed below can cause an inexperienced reader of math to miss a hidden meaning.
"Now” may indicate that new notation or assumptions are about to be introduced. This is often used to begin a new argument.
This use of the word now may have the effect
of canceling
assumptions
made in the preceding text.
"We have shown that if $x\in A$, then $x\in B$. Now suppose $x\notin A$..."
In this text, the author supposes $x\notin A$ presumably because they are doing a proof by cases.
"Now” may be used to point out a fact that is already known or easily deduced and that will be used in the next step of the proof.
In a situation where we already know that $x=7$, one could say, "We have that $x^2+y^2=100$. Now, $x=7$, so $y=\sqrt{5}$." This is similar to the second meaning given for but.
"Now” may simply refer to the point in the text at which it occurs.
"We are now in a position to give a conceptual proof of Pringsheim's Theorem."
The metaphor here is that the reader has been reading straight through the text (unlike a grasshopper) and at the current moment she sees this word "now”. As such it does not really add anything to the meaning.
The Handbook article on "now" includes several more examples.
Atish Bagchi.
A theorem is said to be obvious if the speaker’s mental representation of the mathematical object mentioned in the theorem makes the truth of the fact immediately perceivable. "Obvious" should be used carefully; you need to be sure the listener shares your particular mental representation! See also trivial.
The order of a mathematical object is a number (or other math object) associated with that object. What "order" means depends on the mathematical specialty.
The word partition has several distinct meanings in math.
All of these ideas involve the concept of breaking something up into distinct pieces, but they are genuinely distinct ideas.
Permutation is defined in the literature in two different ways:
Of course, the two definitions can be converted into each other, but they give different mental images. The discussion in Wikipedia gives both definitions.
In most texts and university courses, the phrase " $x$ is positive" means $x \gt 0$. In a few texts and in France (and possibly other places in Europe), it may mean $x\ge 0$.
Here are three variant phrases that say that $125=5^3$:
Some students are confused by such statements, and conclude that $3$ is the "power". This usage appears in print in Wikipedia in its entry on Exponentiation (as it was on 22 November 2016):
...$b^n$ is the product of multiplying $n$ bases: \[b^n = \underbrace{b \times \cdots \times b}_n\] In that case, $b^n$ is called the $n$-th power of $b$, or $b$ raised to the power $n$.
As a result, students (and many mathematicians) refer to $n$ as the "power" in any expression of the form "$a^n$". The number $n$ should be called the "exponent". The word "power" should refer only to the result $a^n$. I know mathematical terminology is pretty chaotic, but it is silly to refer both to $n$ and to $a^n$ as the "power".
Almost as silly as using $(a,b)$ to refer to an open interval, an ordered pair and the GCD. (See The notation $(a,b)$.)
Suggestion for lexicographical research: How widespread does referring to $n$ as the "power" come up in math textbooks or papers? (See usage.)
Thanks to Tomaz Cedilnik for comments on the first version of this entry.
A prime is a typographical symbol, and also a positive integer that has exactly two positive integer divisors (Wi, MW).
This word is used in conflicting ways in mathematical writing and in mathematical logic.
The word quotient is used with many different meanings in math, all of which are closely related. The Wikipedia article covers a lot of these meanings.
The range of a function may mean its codomain or its image. Both usages are common in undergraduate texts.
The property of rank is defined for many different kinds of math structures. The word generally means different things for different structures. The Wikipedia article covers many of these meanings.
Group, abelian group, permutation and permutation group each has a different meaning of "rank", and graph has at least two unrelated meanings of "rank".
Used before giving a mathematical fact (definition, theorem or proof).
"Recall that an integer is even if it is divisible by 2."
The intent seems to be that the author expects that you already know the meaning of the defined term, but just in case here is a polite reminder.
I often see "recall that" when I don't recall anything of the sort.
In the United States, to revise a document means to change it, hopefully improving it in the process. Speakers influenced by British English use "revise" to mean what Americans mean by "review". In particular, students may talk about "revising" for an upcoming test. In this case there is no implication that anything will (or will not) be changed.
If $f(x)={{x}^{2}}$, you may say, "For negative $x$, $f$($x$) and $f$'($x$) are opposite in sign." Observe that for negative $x$, the expression $f'(x)$ denotes a negative number even though the expression has no negative sign in it.
An expression such as ${{a}_{2}}$ commonly means that $a$ is the entry indexed by $2$ in a sequence. Since a sequence is a function on its index set, authors may refer to ${{a}_{i}}$ as a "function on $I$", and use language such as"$a$ is increasing in $I$".
In ordinary English, if you subtract from a collection you make it smaller, and if you add to a collection you make it bigger.
When $a$ and $b$ are numbers, to subtract $b$ from $a$ means to add $-b$ to $a$, written $a-b$. This process can clash with the ordinary English meaning of "subtract".
This difference in the mental image of subtraction can may you think, for example, that $x-y$ must be smaller that $x$. But that is wrong if $y$ is negative!
Both these usages (adding to a collection and applying the operation of addition) occur in mathematical writing.
See also minus and subtraction.
For an assertion $P$, a phrase of the form "$c$ such that $P(c)$" means that the assertion $P$ must be true of $c$.
$P$ is sufficient for $Q$ if the statement "If $P$, then $Q$" is true (see conditional assertion.) You can also say $P$ suffices for $Q$. The idea behind the word is that to know that $Q$ is true it is enough to know that $P$ is true.
As used in math, a symbol, is either
These two meanings are genuinely different. The symbol $\phi$ has two distinct meanings in these sentences:
It is best if an author puts quotes around the symbol when they mean the typographical symbol, but they don't always do that. See Mathworld and Wikipedia for much more about symbols.
Symbol manipulation is the use of algebraic rules, or rules of some other computational system, to change a symbolic expression to an equivalent one. For example, you can change $3{{x}^{2}}+3$ to the equivalent expression $3({{x}^{2}}+1)$ using the distributive law for algebra, and you can change $\int_{1}^{3}{{{x}^{2}}+{{x}^{3\,}}\,dx}$ into $\int_{1}^{3}{{{x}^{2}}\,dx}+\int_{1}^{3}{{{x}^{3}}\,dx}$ using a rule of integration.
The word tangent may refer either
These two meanings are related: the trigonometric tangent of an angle $\theta$ is by definition the length of a certain line segment (red in the picture) tangent to the circle of radius 1. The length of the brown line is $\cos \theta$ and that of the green line is $\sin \theta$, which by similar triangles gives you the equation \[\tan\theta=\frac{\sin\theta}{\cos\theta}\]
The word term is used is several ways in math English.
A term is one constituent of a sum or sequence (finite or infinite), in contrast to a factor, which is a constituent of a product.
The terms in the expression ${{x}^{2}}+2x+1$ are ${{x}^{2}}$, $2x$ and $1$.
This is discussed here. Note that this use and the preceding one conflict. For example, the expression "${{x}^{3}}\left( {{y}^{2}}-1 \right)$" contains symbolic terms ${{x}^{3}}$and ${{y}^{2}}-1$, which would in math English be called factors rather than terms.
"Writing a function in terms of $x$" means giving a defining equation containing $x$ as the only variable (but it may contain parameters).
The English word "the" is called the definite article. It is used in forming definite descriptions: phrases that describe a particular individual.
A set $\left\{ x|P(x) \right\}$ is often described in setbuilder notation with a phrase such as "the set of $x$ such that $P(x)$". This means the set is the set of all $x$ for which $P(x)$ is true.
To call an statement a theorem is to claim that the statement has been proved.
In older mathematical writing, the phrase theory of functions refers by default to the theory of analytic functions of one complex variable.
If you write "$\sin x$" meaning the sine of $x$ degrees, you are not using the same function as when you write "$\sin x$", meaning the sine of $x$ radians. They have different derivatives, for example: Let $\sin x$ denote the sine of $x$ radians and $\text{sind}\ x$ the sine of $x$ degrees, so that $\text{sind}\,x=\sin \frac{\pi }{180}x$. Then $\frac{d\sin x}{dx}=\cos x$ but $\frac{d\,\text{sind}\,x}{dx}=\frac{\pi }{180}\cos x$.
The same remark may be made of the other trig functions. In postcalculus pure mathematics "$\sin x$" nearly always refers to the sine of $x$ radians, often without explicitly noting the fact. This is not true for texts written by non-mathematicians, who at least often use the degree sign to tell you their meaning.
See also logarithm and tangent.
A theorem is said to be trivial to prove or trivially true
Here is a scenario that exemplifies (a):
I have seen this happen several times with this and other theorems. This sort of incident may be why many intelligent people feel they are "bad at math".
People are not born knowing |
This example illustrates (b).
Basic facts about groups and subgroups learned in first semester abstract algebra:
Now if $H$ has index $2$ that means that each partition consists of two cosets. In both cases, one of them has to be $H$, so the other one has to be $G\setminus H$, which must therefore be a left and right coset of $H$. So $H$ is normal in $G$.
So once you understand the basics about cosets and normal subgroups, the fact that $H$ has to be normal if it is of index $2$ is "obvious". I don't think you should call this "trivial". Best to say it is "obvious if you have a clear understanding of cosets of groups".
A mathematical structure is said to be trivial if its underlying set is empty or a singleton set. In particular, a subset of a set is nontrivial if it is nonempty. I have not found an example where "nontrivial subset" means it is not a singleton. See also proper.
The truth value of a statement is "true" if the statement is true and "false" if it is false.
The truth value of "$7 + 3 = 10$" is "true" and the truth value of "$7 + 3 = 11$" is "false".
Assertions with variables in them need not be either true or false.
If $n$ is an integer, the statement "$n \gt 7$" is neither true nor false as long as nothing is known about $n$. If you know $n = 42$, then you know that "$n \gt 7$" is true. If you know that $n \gt 4$, you still don’t know the truth value of "$n \gt 7$". In this situation, some authors say the truth value is indeterminate.
In formal mathematical logic, a statement or an assertion becomes a math object, although this point is not often made clear in logic texts. An assertion with variables is a variable math object whose values range over $\{\text{true},\text{false},\text{indeterminate}\}$. See my post on variables. Atish Bagchi and I have called traditional mathematical logic string-based logic in this paper (which proposes a graph-based logic).
In mathematical discourse, two mathematical objects can be one object! This is because two distinct names for a particular kind of math object usually are allowed to have the same value unless some word such as "distinct" is used to ensure that they are different.
Once upon a time (before October 2006), Wikipedia’s definition of boolean algebra said until recently that a boolean algebra has "…two elements called $0$ and $1$…" It did not say they must be distinct in the axiom but in the examples it said that the two-element boolean algebra is the simplest example.
There is more detail, including citations, in my post "Two".
Under is used to name the function or relation just referred to in the sentence. The reference may be indirect or implicit.
It is my impression that some mathematicians use "under" in this way a lot, but most mathematicians don’t use it at all. This needs more lexicographical research than what I did for the Handbook article on "under"..
A mathematical structure is likely to be defined as a set $S$ along with some other mathematical objects that obey certain axioms. The set $S$ is the underlying set of the structure.
One or more variables may occur in a constraint, and the intent of the discourse may be to determine the values of the variables that satisfy the constraint. In that case the variables may be referred to as unknowns.
In both these problems $x$ would be called an unknown.
A typical definition in mathematics may make use of a number of previously defined concepts. To unpack or unwind such a definition is to replace the defined terms with explicit, spelled-out requirements. See translation problem and rewrite using definitions.
Similarly a function may be defined by a complicated formula. To unpack such a formula means investigating it piece by piece, or chunk by chunk. Zooming and Chunking has an example.
Let $E$ be an equivalence relation. To say that a definition or description of a type of mathematical object determines the object up to $E$ (or modulo $E$) means that any two objects satisfying the description are equivalent with respect to $E$.
The value of the function defined by $f(x)={{x}^{2}}+2$ is $11$. You could also say that the value of the expression $x^2+2$ is $11$.
Don’t misread "value" as meaning "worth". |
A function $f$ vanishes at an input $a$ if $f(a)$ = 0. For example, the function $f(x):=x^3-27$ vanishes at $3$. This (pretty common) usage can cause cognitive dissonance, since the function still exists!
The word vector has (at least) three different useful mental representations:
Of course, the third representation includes the other two, but with some subtleties. For example, to think of an element of an abstract $n$-dimensional vector space as a $n$-tuple requires choosing a basis for the space. There is in general no canonical choice of basis.
In computer engineering, the word vector is often used to refer to an $n$-tuple of any sort of thing, not necessarily elements of a field (or numbers at all), so that the $n$-tuple may indeed not be a member of a vector space. I have heard this usage in conversation but could not find an unequivocal citation for it for the Handbook.
Suppose a function $F$ is defined by an assertion "$F(X) = E$", where $X$ and $E$ are some expressions in the symbolic language. $F$ is said to be well-defined if:
Let $F:\mathbb{R}\to\mathbb{R}$ be the function defined by $F(|x|)=x^2$. Then $F$ is well defined because $|x|=|y|$ if and only if $x=y$ or $x=-y$, so we can check that $F(|x|)=x^2$ and $F(|-x|)=(-x)^2=x^2$. Note that this does not work for $G(|x|)=x^3$ because then $G(|-1|)=(-1)^3=-1$ but $G(|1|)=1^3=1$.
"When a function has a derivative, it is necessarily continuous."
These phrases
all mean if and only if.
"A function has a derivative precisely when it is continuous" is false: for example, the absolute value function is continuous but doesn't have a derivative at $0$.
Whenever is also used to mean "if". It may be used after another "if" to avoid having two "if"s in a row, for example
"A relation $\alpha $ is symmetric if whenever $x\alpha y$, then $y\alpha x$".
Where may be used in a special way in math English, sometimes without any connotation of location.
A proof of an assertion involving two elements $x$ and $y$ of some mathematical structure $S$ might appear to require consideration of two cases in which $x$ and $y$ are related in different ways to each other. For example for some assertion $P$, $P(x, y)$ or $P(y, x)$ could hold. However, if there is a symmetry of $S$ that interchanges $x$ and $y$ without affecting the property in question, you may need to consider only one case. In that case, the proof may begin with a remark such as,
"Without loss of generality, we may assume $P(x, y)$."
WLOG is an abbreviation for this phrase.
Let $r$ and $s$ be real numbers. The mean (or average) of $r$ and $s$ is $\frac{r+s}{2}$. You would expect that the average is between $r$ and $s$ in value. Here is a formal proof that $\frac{r+s}{2}$ is between $r$ and $s$:
Assume without loss of generality that $r\lt s$. Then $\frac{r}{2}\lt\frac{s}{2}$, so \[r=\frac{r+r}{2}\lt\frac{r+s}{2}\lt\frac{s+s}{2}=s\]
The reason you can assert WLOG in this case is that if $r\gt s$ then you can simply switch their names. You can do this because if a number is between $r$ and $s$ then it is between $s$ and $r$. That property of betweenness is the symmetry referred to in the definition of WLOG above.
A witness to an existential statement of the form $\exists x\,(P(x))$ is an object $w$ for which $P$($w$) is true. For example, a witness to the statement "There is an integer $x$ for which ${{x}^{2}}\gt 5$" is the integer $3$ (among many others). The integer $2$ is not a witness to this statement because ${{2}^{2}}=4$, which is not greater than $5$. On the other hand, the statement "There is an integer $n$ for which $n\gt n$" is false, so it has no witnesses at all.
The number zero is an integer. It is the number of elements in the empty set. In American college and university usage it is neither positive nor negative, but many college students show confusion about this.
There is more about zero in the abmath entry for the empty set symbol.
A zero of a function $f(x)$ of one variable is an element $c$ the domain of $f$ for which $f(c) = 0$. It is common to use the word root for such an element $c$.
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