This page is here to help review notation and terminology for those new to, or requiring a refresher, of sets and set theory.

# Set NotationΒΆ

Recall that a set is a collection of distinct objects. In any given set each object in
the set is called an **element** of the set.

For example, if \(S\) represents a set, and \(a\) is an element of set \(S\), we denote this by writing

\[a \in S.\]

If \(a\) *is not* an element of set \(S\), we write it as

\[a \not\in S.\]

We can more explicitly write out what the specific elements of a set are by using braces: \(\{,\}\). For example, if I wanted \(S\) to be the set of integers between 2 and 6 inclusively, then I can write

\[S = \{2,3,4,5,6\}.\]

There is also the set with no elements at all in it, and it is called the **empty set**, and is
denoted by \(\emptyset\). Other ways of denoting are by using \(\varnothing\), or even
\(\{\}\).

We also use colons to represent conditions on elements in a particular set. This lets us build up more complicated sets. For example, if I use \(\mathbb{Z}\) to represent the set of all integers, then if I want \(T\) to represent the set of all integers greater than five, I can write it as

\[T = \{n \in \mathbb{Z} : n > 5\}.\]

Colons in combination with braces and \(\in\) is called **set builder notation** and lets us
create very complicated sets. Let’s say I want to create the set of all irrational numbers, then
I can do this by starting with the sets of real numbers, rational numbers, and using set
builder notation:

Let \(\mathbb{R}\) represent the set of real numbers, and \(\mathbb{Q}\) the set of rational numbers, then if we use \(X\) to represent the set of irrational numbers, we can write it as

\[X = \{ r \in \mathbb{R} : r \not\in \mathbb{Q} \}.\]

We can add even more conditions. Let us use \(Y\) to represent the set of negative rational numbers, then we can write it as

\[Y = \{ r \in\mathbb{R} : r\not\in\mathbb{Q}\ \&\ r<0 \},\]

or alternatively by going back to \(X\) and saying

\[Y = \{ x \in X : x<0 \}.\]

To help write conditions precisely without creating too much clutter, we will sometimes use symbols to represent logical quantifiers such as \(\exists\) to say “there exists” or \(\forall\) to represent “for all”. For example, if we wanted to say that a set \(S\) was a subset of a set \(T\), that is, any element of \(S\) is also an element of \(T\), we can write that as

\[\forall x, x \in S \implies x \in T.\]

See also set notation and set (mathematics) for more information.