set

A set is a collection of objects, called elements or members of the set.

Formally, a set $S$ is constructed

where $s_i$ are the elements of $S$.

If an element $a$ is a member of set $A$, we say $a\in A$.

Sets are uniquely characterized by their elements; two sets that have precisely the same elements are equal as sets (they are the same set). In defining a set, ordering of elements does not matter.

The number of elements in a set is its cardinality.
There is a unique set with no elements called the empty set (written $\varnothing$).
A set may be finite or infinite.

In set-builder notation, a set $S$ is constructed

where the colon is read as "such that", "for which", or "with the property that";
the formula $\mathrm{\Phi }(x)$ is the indicator function, where all values of $x$ for which $\mathrm{\Phi }(x)=$true have set membership, and any value of $x$ for which $\mathrm{\Phi }(x)$=false does not have set membership.

In general, it is good practice to establish a universe that serves as the domain for our indicator function. However, it is also good practice to specify the domain explicitly when defining a set.

We typically write, for a domain $E$,

which is equivalent to

An extension of set-builder notation replaces the single variable $x$ with a formula $f(x)$:

which should be read as

i.e. "the set of values $y$ output by $f(x)$ for each input $x$ which satisfies $\mathrm{\Phi }(x)$"

Example 1

$\{2n:n\in \mathbb{N}\}$ is the set of all even natural numbers

Example 2

$\mathbb{Q}=\{\frac{p}{q}:p,q\in \mathbb{Z},q\ne 0\}$ is the set of rational numbers