Produced by Charles Wells Revised 2017-01-27 Introduction to this website website TOC website index blog Back to head of understanding math chapter Back to Variables and substitution

In many mathematical texts, the variable $x$ may denote a real number, although which real number may not be specified. This is an example of a variable mathematical object. This point of view and terminology is not widespread, but I think it is worth understanding because it provides a deeper understanding of some aspects about how math is done.

It is useful to distinguish between ** specific ** math objects and ** variable ** math objects.

- The number $42$ (the math object represented as "42" in base $10$, "2A" in hexadecimal and "XLII" as a Roman numeral) is a
**specific**math object. It is not any of the representations just listed -- they are just strings of letters and numbers. - The ordered pair $(4,3)$ is a specific math object. It is not the same as the ordered pair $(7,-2)$, which is
*another*specific math object. - The sine function $\sin:\mathbb{R}\to\mathbb{R}$ is a specific math object. You may know that the sine function is also defined for all complex numbers, which gives
*another*specific math object $\sin:\mathbb{C}\to\mathbb{C}$. - The group of symmetries of a square is a specific math object. (If you don't know much about groups, the link gives a detailed description of this particular group.)

Math books are full of references to math objects, typically named by a letter or a name, that are not completely specified. Some mathematicians call these **variable objects** (not standard terminology). The idea of a variable mathematical object is not often taught as such in undergraduate classes but it is worth pondering. It has certainly clarified *my* thinking about expressions with variables.

- If an author or lecturer says
"Let $x$ be a real variable", you can then think of $x$ as a
**variable real number**. In a proof you can't assume that $x$ is any*particular*real number such as $42$ or $\pi$. - If a lecturer says, "Let $(a,b)$ be an ordered pair of integers", then all you know is that $a$ and $b$ are integers. This makes $(a,b)$ a
**variable ordered pair**, specifically a pair of integers. The lecturer will not say it is a variable ordered pair since that terminology is not widely used. You have to understand that the phrase "Let $(a,b)$ be an ordered pair of integers"*implies*that it is a variable ordered pair just because "a" and "b" are letters instead of numbers. - If you are going to prove a theorem about functions, you
might begin, "Let $f$ be a continuous function", and in the
proof refer to $f$ and various objects connected to $f$. This
makes $f$ a
**variable mathematical object**. When you are proving things about $f$ you may use the fact that it is continuous. But you cannot assume that it is (for example) the sine function or any other*particular*function. - If someone says, "Let $G$ be a group" you can
think of $G$ as a
**variable group**. If you want to prove something about $G$ you are free to use the definition of "group" and any theorems you know of that apply to all groups, but you can't assume that $G$ is any specific group.

A logician would refer to the symbol $f$, thought of as denoting a function, as a **variable**, and likewise the symbol $G$, thought of as denoting a group. But mathematicians in general would not use the word "variable" in those situations.

The idea that $x$ is a variable object means thinking of $x$ as a genuine mathematical object, but with limitations about what you can say or think about it. Specifically,

Some assertions about a variable math object

may be neither true nor false.

The statement, "Let $x$ be a real
number" means that $x$ is to be regarded as a **variable real number** (usually called a "real variable"). Then you know the following facts:

- The statement "${{x}^{2}}$ is
not negative" is
*true*. - The assertion "$x=x+1$"
is
*false.* - The assertion "$x\gt 0$" is
*neither true nor false.*

Suppose you are told that $x$ is a real number and that ${{x}^{2}}-5x=-6$.

- You know (because it is given) that the statement "${{x}^{2}}-5x=-6$" is
*true*. - By doing some algebra, you can discover that the statement "$x=2$ or $x=3$" is
*true*. - The statement "$x=2$
*and*$x=3$" is*false*, because $2\neq3$. - The statement "$x=2$" is
*neither true nor false*, and similarly for "$x=3$". - This situation could be described this way: $x$ is a variable real number varying over the set $\{2,3\}$.

This example may not be easy to understand. It is intended to raise your consciousness.

A prime pair is an ordered pair of integers $(n,n+2)$ with the property that both $n$ and $n+2$ are prime numbers.

Definition: $S$ is a PP set if $S$ is a set of pairs of integers with the property that every pair is a prime pair.

- "$\{(3,5),(11,13)\}$ is a PP set" is
*true*. - "$\{(5,7),(111,113),(149,151)\}$ is a PP set" is
*false*.

Now suppose $SS$ is a **variable PP set**.

- "$SS$ is a set" is
**true**by definition. - "$SS$ contains $(7,9)$" is
**false**. - "$SS$ contains $(3,5)$" is
**neither true nor false**, as the examples just above show. - "$SS$ is an infinite set":
- This is certainly not true (see finite examples above).
- This claim may be neither true nor false, or it may be plain false, because no one knows whether there is an infinite number of prime pairs.
- The point of this example is to show that "we don't know" doesn't mean the same thing as "neither true nor false".

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