Sets and Set-Builder Notation

Collections as Objects

A set is a well-defined collection of distinct objects, its elements. Sets are the fundamental objects of modern mathematics — nearly everything else is defined in terms of them. We write $x \in A$ for membership and $x \notin A$ for non-membership. A set is determined entirely by its elements: order and repetition are irrelevant, so $\{1, 2, 2, 3\}$ is the same set as $\{1, 2, 3\}$ and as $\{3, 1, 2\}$.

Sets can be specified two ways. Roster notation lists the elements: $\{2, 4, 6, 8\}$. Set-builder notation states a defining property: $\{x \in \mathbb{Z} : x \text{ is even and } 0 < x < 10\}$, read "the set of integers $x$ such that...". Set-builder is essential for infinite or complex sets that cannot be listed.

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

• $A \subseteq B$ (subset) — every element of $A$ is in $B$.
• The empty set $\emptyset$ has no elements; it is a subset of every set.
• The cardinality $|A|$ is the number of elements.
• The power set $\mathcal{P}(A)$ is the set of all subsets of $A$; a set with $n$ elements has $2^n$ subsets.

Notice the layering: elements belong to sets, sets can be elements of other sets (the power set), and this careful distinction between $\in$ (membership) and $\subseteq$ (subset) is where precision matters most.

Common pitfall: confusing membership ($\in$) with the subset relation ($\subseteq$). "$3 \in \{1,2,3\}$" is true ($3$ is an element), but "$3 \subseteq \{1,2,3\}$" is nonsense — $3$ is not a set. Meanwhile "$\{3\} \subseteq \{1,2,3\}$" is true (the set $\{3\}$ is a subset) but "$\{3\} \in \{1,2,3\}$" is false. An element is in a set; a subset is contained in a set.

A 3-element set with its 8 subsets drawn as a small lattice, the empty set at the bottom and the full set at the top in accent — the power set of size 2^3.

$$|\mathcal{P}(A)| = 2^{|A|}$$