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A set is a well-defined collection of distinct objects. Membership (x ∈ S) and subset relationships (A ⊆ B) form the foundation for formal mathematics. The principle of extensionality—two sets are equal if and only if they have the same elements—provides the rigorous framework for reasoning about sets throughout mathematics.
You have already worked with predicates and quantifiers — statements like "x is even" that are true or false depending on the value of x. Set theory formalizes the next step: collecting all objects that satisfy a predicate into a single mathematical entity. The set of all even integers, for example, is {x : x is even}. This notation, called set-builder notation, directly connects sets to the predicate logic you already know.
Two features distinguish sets from informal collections. First, sets contain only *distinct* elements — writing {2, 2, 5} is the same as writing {2, 5}. Repetition carries no meaning. Second, sets are *unordered* — {1, 2, 3} and {3, 1, 2} are identical. What matters is membership, not position. The principle of extensionality captures both ideas: two sets are equal if and only if every element of one is an element of the other, and vice versa. This gives mathematics a precise definition of equality for collections.
The membership relation x ∈ S is the atomic building block of set theory. From it, we define the subset relation: A ⊆ B means that for every x, if x ∈ A then x ∈ B. Notice this is just a universally quantified conditional — the logic you already know. A proper subset (A ⊂ B) adds the condition that A ≠ B, so B contains at least one element not in A. The empty set ∅ is a subset of every set, because the statement "for every x, if x ∈ ∅ then x ∈ B" is vacuously true — there is nothing in ∅ to check.
These definitions might seem overly careful, but the precision pays off immediately when you start doing proofs. To prove A ⊆ B, you take an arbitrary element x ∈ A and show it must also be in B — a direct application of the universal quantifier proof strategy. To prove two sets are equal, you prove A ⊆ B and B ⊆ A, which is extensionality in action. Every proof about sets reduces to the membership relation and the logic you already mastered.