Sets and Their Representations

A set is a well-defined collection of distinct objects. "Well-defined" means that,

given any object, we can say with certainty whether it belongs to the collection or

not. For example, "the collection of all vowels in the English alphabet" is a set,

because we can decide unambiguously whether any given letter is a vowel. On the other

hand, "the collection of intelligent students in a class" is not a set in the

mathematical sense, because "intelligent" is not a well-defined criterion — different

people may disagree on who qualifies.

Sets are usually denoted by capital letters $A, B, C, \dots$ and their elements (or

members) by lowercase letters $a, b, c, \dots$. 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 of $A$, we write

$x \notin A$.

Standard number sets. Certain sets of numbers occur so often that they have their

own reserved symbols:

Representing a set. There are two standard methods.

1. Roster (or tabular) form. All the elements of the set are listed, separated by

   commas, and enclosed within braces $\{\ \}$. The order of listing does not matter,

   and no element is listed more than once. For example, the set of natural numbers

   less than 6 is written as $\{1, 2, 3, 4, 5\}$.

2. Set-builder (or rule) form. Instead of listing every element, we state a

   property (a rule) that every element of the set — and no other object — must

   satisfy. It is written as $\{x : p(x)\}$ or $\{x \mid p(x)\}$, read "the set of all

   $x$ such that $x$ has the property $p(x)$." For example,

Worked illustration. The set $B = \{2, 4, 6, 8, 10\}$ can be written in

set-builder form as $B = \{x : x = 2n,\ n \in N,\ 1 \le n \le 5\}$, or equivalently

$B = \{x : x \text{ is an even natural number}, x \le 10\}$. Both descriptions pin

down exactly the same five elements, which is why roster form and set-builder form

are interchangeable — one lists, the other characterizes.

Cardinality. If a set $A$ has a finite number of distinct elements, that number is

called the cardinal number of $A$, written $n(A)$. For $A = \{1, 2, 3, 4, 5\}$,

$n(A) = 5$.