Sets and Their Representation

Much of the mathematics used in commerce and statistics — grouping customers by region, classifying products, describing the set of possible outcomes of a business decision — begins with the simple idea of a collection. A set is a well-defined collection of distinct objects. "Well-defined" is the key phrase: given any object, it must be possible to decide, without any doubt or personal opinion, whether that object belongs to the collection or not. "The set of the first five natural numbers" is well-defined; "the set of good investments" is not, because whether an investment is "good" is a matter of opinion.

The objects in a set are called its elements (or members). Sets are usually named with capital letters $A, B, C, \dots$ and elements with small letters. If $x$ is an element of a set $A$ we write $x \in A$ (read "$x$ belongs to $A$"); if $x$ is not an element we write $x \notin A$. For example, if $A$ is the set of vowels in the English alphabet, then $a \in A$ but $b \notin A$.

Two features of a set follow straight from the definition. First, the elements are distinct — an element is either in the set or not, so we never list it twice; the collection $\{2, 2, 3\}$ is just the set $\{2, 3\}$. Second, order does not matter — $\{2, 3\}$ and $\{3, 2\}$ are the same set, because they have exactly the same members.

Two ways to describe a set. A set can be written in two standard forms:

• Roster (or listing) form: all the elements are listed, separated by commas, inside curly braces $\{\ \}$. For example, the set of odd numbers less than $10$ is $\{1, 3, 5, 7, 9\}$. For a set with a clear pattern that continues, three dots (an ellipsis) show the continuation, e.g. the natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$.
• Set-builder (or rule) form: instead of listing, we state the property that every element satisfies, in the form $\{x : P(x)\}$ or $\{x \mid P(x)\}$, read "the set of all $x$ such that $x$ has property $P$". For example, $\{x : x \text{ is an odd natural number and } x < 10\}$ describes the same set $\{1, 3, 5, 7, 9\}$.

Set-builder form is especially useful when a set is infinite or when listing every element would be impractical — for instance, $\{x : x \in \mathbb{R},\ 2 \le x \le 5\}$ describes every real number from $2$ to $5$, which cannot be listed one by one.

Standard number sets are used so often that they have fixed symbols: $\mathbb{N}$ = natural numbers $\{1, 2, 3, \dots\}$; $\mathbb{W}$ = whole numbers $\{0, 1, 2, 3, \dots\}$; $\mathbb{Z}$ = integers $\{\dots, -2, -1, 0, 1, 2, \dots\}$; $\mathbb{Q}$ = rational numbers (all numbers of the form $\dfrac{p}{q}$ with $p, q \in \mathbb{Z}$ and $q \ne 0$); and $\mathbb{R}$ = real numbers.