Produced by Charles Wells Revised 2017-03-03 Introduction to this website website TOC website index blog back to head of sets chapter

The *definition* of set can be stated in several different ways, all of them complicated. The most widely used definition is based on the Zermelo-Fraenkel axioms.

It is not necessary to understand or even know the Zermelo-Fraenkel axioms to understand sets as they are used in undergraduate math or large parts of graduate math. Nearly everything having to do with sets in ordinary mathematical practice derives from the Method of Comprehension, so there is usually no need for the axiomatic definition. In that sense, the following specification contains everything you need to know about sets for *most* mathematical purposes.

The concept of specification of a math object is discussed in the chapter on Definitions.

A set is a single math object distinct from but completely determined by what its elements are.

This specification tells you the *operative properties* of a set rather than giving a definition in terms of previously known objects.

The specification just given is not a mathematical definition. In particular, in some situations that usually do not occur in most branches of math, a bunch of elements may not correspond to a set. One such example is this:

There is no "set of all sets".

In other words, you can't have a set that is completely determined by the fact that its elements are all the sets that exist. This follows from Cantor's Theorem. In most cases, if you think up a bunch of elements, they do form a set.

The important thing to understand is what the specification means in practice. That is the subject of the next section:

- A set is not merely a typographically convenient way to define a certain collection of things. A set is a math object, just like the number $143$ and the sine function and the real line.
- The numbers $3$ and $42$ are two different things. The set $\{3,42\}$ is
**one thing**.

If someone defines $S$ as the set of all integers bigger than $3$, then the spec means you know all these things:

- $4$ and $10^{42}$ are elements of $S$.
- $3$ and $-99$ are not elements of $S$.
- $S$ is not any of the integers bigger than $3$, because $S$ is a set and an integer is not a set.
- $S$ is not all the integers bigger than $3$ because $S$ is just one thing.

In list notation, the order in which you list the elements of a set is irrelevant for the purposes of determining what the set is. This follows directly from the specification.

- The set
$\{1, 2, 4, 5\}$ by definition contains the numbers $1$, $2$, $4$ and $5$ and
*nothing else.* - So does the set $\{1,5,4,2\}$. The presence of "$5$" in the expression "$\{1,5,4,2\}$" tells you that $5$ is in the set. It does not matter if you say $5$ is in the set before or after you say $4$ is in the set.
- Since a
set is “completely determined by what its elements are”, the expressions "$\{1, 2, 4, 5\}$" and
"$\{1,5,4,2\}$" denote the
*same set*. - Note that the expressions "$\{1, 2, 4, 5\}$" and
"$\{1,5,4,2\}$" themselves are two
*different*things. Compare: "$16$" and "XVI" are two different expressions denoting the same number.

The list notation $\{3, 3, 4\}$ defines a set with two elements $3$ and $4$. The first occurrence of ‘$3$’ in the list says that $3$ is in the set. The second occurrence says the same thing. Saying a true thing twice has no effect (except to irritate the reader). So repetition in list notation does not matter.

Warning: Mathematica uses curly brackets to denote *lists*, not sets. So in Mathematica, $\{1, 2, 4, 5\}$ and
$\{1,5,4,2\}$ are two different lists and so are $\{3,4\}$ and $\{3,3,4\}$.

If $A$ and $B$ are sets, then $A=B$ if and only if $A$ and $B$ have the same elements. In other words:

$A = B$ if and only
if every element of $A$ is an element of $B$

and every element
of $B$ is an element of $A$.

- For
real numbers
*x*, \[\left\{ x|{{x}^{2}}=1 \right\}=\left\{ 1,\,\,-1 \right\}\] (see setbuilder notation) because (a) $1$ and $–1$ satisfy the equation ${{x}^{2}}=1$ and (b) no other real number satisfies that equation. - For real numbers $x$, \[\left\{ x|{{x}^{2}}=2 \right\}\ne \left\{ \sqrt{2} \right\}.\]It is true that $\sqrt{2}$ satisfies the equation ${{x}^{2}}=2$, but it is also true that $-\sqrt{2}$ satisfies ${{x}^{2}}=2$. Since $-\sqrt{2}$ is not listed as an element of $\left\{ \sqrt{2} \right\}$, $\left\{ \sqrt{2} \right\}$ is not equal to $\left\{ x|{{x}^{2}}=2 \right\}$.
- But of course, \[\left\{ x|{{x}^{2}}=2 \right\}=\left\{ \sqrt{2},-\sqrt{2}\right\}.\]

A set, being a math object, can be an element of another set. Furthermore, if it is, its elements are not necessarily elements of that other set because the specification says that a set is a math object that is distinct from its elements.

Let $A= \{ \{1, 2\}, \{3\}, 1, 6\}$.

- $A$ has four elements, two of which are sets.
- $2\in \left\{ 1,\,\,2 \right\}$ and $\left\{ 1,\,\,2 \right\}\in A$, but $2\notin A$. The set $\{1,2\}$ is distinct from its elements, so that even though one of its elements is $2$, the set $\{1,2\}$ itself is not $2$.
- On the
other hand, $1$
*is*an element of $A$ because it is explicitly listed as such.

Let \[B=\left\{ \left\{ 1 \right\},\,\left\{ 2
\right\},\,\left\{ 1,\,2 \right\},\,\varnothing \right\}.\]Then $B$ is the
**set of all subsets** of the set $\{1, 2\}$. In particular, $\varnothing \in B$ (the
empty set is an element of $B$.) Note that the empty set is *not* an element of the set $A$ of the preceding example.

It is a myth that the empty set is an element of every set.

On the other hand:

The empty set is a subset of every set.

Most of the time in practice either *none* of
the elements of a set are sets or *all* of them are. In fact,
sets such as $A$ and $B$ in the preceding examples, which have
both sets and numbers as elements, rarely occur in mathematical writing
except as examples in texts such as the one you are reading which are intended to bring out the
difference between "element of" and "included in". See
also contain.

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