1.1 Describing a Set

Set: A set is a collection of objects.

Notation: Let \(A\) be a set. If \(a\) is an element of \(A\), we write \(a \in A\). Otherwise, \(a \notin A\).

The Empty Set: The empty set \(\emptyset\) contains no elements.

Describing a set

Explicit list: \(S = \{ 1, 2, 3 \}\)
By a property: \(S = \{ x \in \mathbb{U} : p(x) \}\)
- Read: all \(x\) in the universal set \(\mathbb{U}\) such that \(x\) satisfies some property \(p\). \[ \begin{align} S &= \{ x \in \mathbb{R} : x^2 + x - 1 = 0 \}\\ T &= \{ x \in \mathbb{Z} : \left|{x - 2}\right| = 5 \} \end{align} \]

Notation: \(|S|\) is the cardinality of, or number of elements in, \(S\).

\[ \begin{align} |\emptyset| &= 0\\ \left|\{ 1, \{ \emptyset, 15 \} \}\right| &= 2 \end{align} \]

Special Sets