Sets are collections of distinct objects. Membership (x ∈ S), subset (A ⊆ B), union (A ∪ B), intersection (A ∩ B), complement (A'), and Cartesian product (A × B) are fundamental operations. Set-builder notation {x : P(x)} describes sets by properties. Understanding set operations is essential for formalizing mathematical definitions.
Work with small concrete sets and visualizations (Venn diagrams). Practice converting between set-builder and roster notation.
From your work with the universal quantifier, you know that mathematical statements like "for all x in S, P(x)" rely on having a precise notion of what S is. Sets formalize this: a set is an unordered collection of distinct objects, where membership is the foundational relation. Writing x ∈ S means "x is a member of S" — a binary, yes-or-no claim. The empty set ∅ = {} contains no elements and is a subset of every set by vacuous truth: the statement "for all x ∈ ∅, P(x)" is true regardless of P because there are no elements to check.
The subset relation A ⊆ B means every member of A is also a member of B: for all x, if x ∈ A then x ∈ B. This is containment, not equality. A = B if and only if A ⊆ B and B ⊆ A — this double-containment argument is the standard proof strategy for set equality. The most persistent confusion is between ∈ and ⊆: if A = {1, 2, 3}, then 2 ∈ A but {2} ⊆ A. The element 2 is a number; {2} is a set containing a number. They are different kinds of objects, and conflating them breaks formal proofs.
The union A ∪ B = {x : x ∈ A or x ∈ B} collects everything in either set. The intersection A ∩ B = {x : x ∈ A and x ∈ B} keeps only what both sets share. The complement Aᶜ relative to a universal set U is {x ∈ U : x ∉ A}. De Morgan's laws govern how complement distributes over union and intersection: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)᷊ = Aᶜ ∪ Bᶜ. These mirror the logical De Morgan's laws for "or" and "and," which is not a coincidence — set operations are logical operations applied to membership predicates. Verifying De Morgan's laws by Venn diagram first, then by chasing element membership formally, is the best way to internalize them.
Set-builder notation {x ∈ S : P(x)} describes a set by a defining property rather than listing elements. For example, {n ∈ ℤ : n is even} is the set of even integers. This notation directly echoes the universal quantifier: claiming a ∈ {x ∈ S : P(x)} is equivalent to claiming a ∈ S and P(a) holds. The Cartesian product A × B = {(a, b) : a ∈ A, b ∈ B} forms ordered pairs from two sets — the set-theoretic foundation for functions and relations. Because ordered pairs are involved, (a, b) ≠ (b, a) in general, and A × B ≠ B × A unless A = B. These basic operations are the vocabulary you will use to define functions, prove properties of maps, and eventually characterize equivalence relations and partitions.