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Produced by Charles Wells     Revised 2017-02-24
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SETS: NOTATION

This section describes customary usage for defining specific sets.

Element notation

The expression "$x\in A$" means that $x$ is an element of the set $A$.

The expression "$x\notin A$" means that $x$ is not an element of $A$.

Examples

Pronunciation

The assertion "$x\in A$" is pronounced in any of the following ways:

Remarks

Acknowledgment: Atish Bagchi

List notation

Definition: list notation

A set with a small number of elements may be denoted by listing the elements inside braces (curly brackets). The list must include exactly all of the elements of the set and nothing else.

Example

The set $\{1,\,3,\,\pi \}$ contains the numbers $1$, $3$ and $\pi $ as elements, and no others. So $3\in \{1,3,\pi \}$ but $-3\notin \{1,\,3,\,\pi \}$.

Properties of list notation

List notation shows every element and nothing else

If $a$ occurs in a list notation, then $a$ is in the set the notation defines.  If it does not occur, then it is not in the set.

When I say "$a$ occurs" I don't mean it necessarily occurs using that name. For example, $3\in\{2+5,2+4,1+2\}$ but $3\notin\{2+5,2+4,1+6\}$.

The order in which the elements are listed is irrelevant

For example, $\{2,5,6\}$ and $\{5,2,6\}$ are the same set.

Repetitions don't matter

Multisets may be written with braces and repeated entries, but then the repetitions mean something.

When elements are sets

When (some of) the elements in list notation are themselves sets (more about that here), care is required.  For example, the numbers $1$ and $2$  are not elements of the set \[S:=\left\{ \left\{ 1,\,2,\,3 \right\},\,\,\left\{ 3,\,4 \right\},\,3,\,4 \right\}\]The elements listed in $S$ include the set $\{1, 2, 3\}$ among others, but not the number $2$.  The set $S$ contains four elements, two sets and two numbers. 

Another way of saying this is that the element relation is not transitive: The facts that $A\in B$ and $B\in C$ do not imply that $A\in C$. 

Sets are arbitrary

Examples

Setbuilder notation

Definition:

Suppose $P$ is an assertion. Then the expression "$\left\{x\,|\,P(x) \right\}$" denotes the set of all objects $x$ for which $P(x)$ is true. It contains no other elements.

Pronunciation

Examples

In these examples, $n$ is an integer variable and $x$ is a real variable..

Usage and terminology

Setbuilder notation is tricky

Looking different doesn't mean they are different.

A set can be expressed in many different ways in setbuilder notation. For example, $\left\{ x\,|\,x\gt6 \right\}=\left\{ x\,|\,x\ge 6\text{ and }x\ne 6 \right\}$. Those two expressions denote exactly the same set. (But $\left\{x\,|\,x^2\gt36 \right\}$ is a different set!)

Russell's Paradox

In certain areas of math research, setbuilder notation can go seriously wrong. See Russell’s Paradox if you are curious.

This glitch rarely causes problems in undergraduate math courses.

Variations on setbuilder notation

An expression may be used left of the vertical line in setbuilder notation, instead of a single variable.

Giving the type of the variable

You can use an expression on the left side of setbuilder notation to indicate the type of the variable.

Example

The unit interval $I$ could be defined as \[\mathbb{I}=\left\{x\in \mathrm{R}\,|\,0\le x\le 1 \right\}\]making it clear that it is a set of real numbers rather than, say rational numbers.  You can always get rid of the type expression to the left of the vertical line by complicating the defining condition, like this:\[\mathbb{I}=\left\{ x\,|\,x\in \mathrm{R}\text{ and }0\le x\le 1 \right\}\]

Other expressions on the left side

Other kinds of expressions occur before the vertical line in setbuilder notation as well.

Example

The set\[\left\{ {{n}^{2}}\,|\,n\in \mathbb{Z} \right\}\]consists of all the squares of integers; in other words its elements are 0,1,4,9,16,….  This definition could be rewritten as $\left\{m\,|\,\text{ there is an }n\in \mathrm{}\text{ such that }m={{n}^{2}} \right\}$.

Example

Let $A=\left\{1,3,6 \right\}$.  Then $\left\{ n-2\,|\,n\in A\right\}=\left\{ -1,1,4 \right\}$.

Warning

Be careful when you read such expressions.

Example

The integer $9$ is an element of the set \[\left\{{{n}^{2}}\,|\,n\in \text{ Z and }n\ne 3 \right\}\]It is true that $9={{3}^{2}}$ and that $3$ is excluded by the defining condition, but it is also true that $9={{(-3)}^{2}}$ and $-3$ is not an integer ruled out by the defining condition!

References

Set notation. Blog post.

Sets. Blog post.

Acknowledgment

Toby Bartels for corrections.



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