Sets and Set-Builder Notation

Sets and Elements

A set is a well-defined collection of distinct objects called elements. Nearly all modern mathematics is built on sets.

A set is determined entirely by its elements. Order and repetition are irrelevant, so $\{1, 2, 2, 3\} = \{1, 2, 3\}$.

Sets are specified in two ways:

• Roster notation lists elements directly: $\{2, 4, 6, 8\}$.
• Set-builder notation uses a rule: $\{x \in \mathbb{Z} : \text{even and } 0 < x < 10\}$.

Standard number sets have reserved symbols: $\mathbb{N}$ (naturals), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals), and $\mathbb{R}$ (reals).

Subsets and Power Sets

Key relationships define how sets interact with each other and their own sizes:

• Subset ($A \subseteq B$): every element of $A$ is also in $B$. The empty set $\emptyset$ is a subset of every set.
• Cardinality ($|A|$): the number of elements in $A$.
• Power set ($\mathcal{P}(A)$): the set of all subsets of $A$. If $|A| = n$, then the cardinality is given by $|\mathcal{P}(A)| = 2^{|A|}$.

Pitfall: Do not confuse membership ($\in$) with the subset relation ($\subseteq$).

$3 \in \{1,2,3\}$ is true because $3$ is an element. However, $\{3\} \subseteq \{1,2,3\}$ is true because $\{3\}$ is a set. Writing $3 \subseteq \{1,2,3\}$ is nonsense because $3$ is not a set.