4 Sets
Ask a mathematician what a number is, and eventually the answer bottoms out in sets. Ask what a function is, a relation, a vector space, a probability — same answer. Sets aren’t mathematically interesting on their own; almost everything else on this site is, formally, built out of them. That makes them a strange kind of foundation: nearly universal in the formal graph, almost invisible in the conceptual one — you don’t need to think about sets to use a vector space, even though a vector space is, formally, a particular kind of set.
4.1 The intuition
A set is a collection of distinct objects, with no notion of order or repetition — \(\{1, 2, 3\}\) and \(\{3, 1, 2\}\) are the same set, and \(\{1, 1, 2\}\) is just \(\{1, 2\}\). The objects in a set are its elements; \(x \in S\) means “\(x\) is an element of \(S\).” That’s essentially the whole intuition, and it’s deliberately thin — the power of the definition is how little it commits to, which is exactly what lets nearly anything (numbers, functions, other sets) become an element of one.
Standard notation is worth having fluent, since it recurs everywhere from here on: \(\{x : P(x)\}\) (“the set of all \(x\) such that \(P(x)\) holds”) builds a set from a defining property, e.g. \(\{n \in \mathbb{Z} : n > 0\}\) is the positive integers. \(\varnothing\) is the empty set, the unique set with no elements. \(A \subseteq B\) means every element of \(A\) is an element of \(B\).
4.2 A collection that broke its own definition
“A collection of objects satisfying any property you like” sounds harmless. It isn’t. In 1901, Bertrand Russell noticed that the property “\(S \notin S\)” (a set that does not contain itself) defines a set — \(R = \{S : S \notin S\}\) — and then asked the natural question: is \(R \in R\)? If yes, then by \(R\)’s own definition \(R \notin R\). If no, then \(R\) satisfies the defining property, so \(R \in R\). Either answer contradicts itself. This is Russell’s paradox, and it’s not a curiosity — it showed that “any property defines a set” is an inconsistent foundation, full stop, and it’s exactly the kind of failure that forces the move from an informal definition to an axiomatic one described in Axioms and Definitions.
The fix mathematicians settled on — the Zermelo–Fraenkel axioms (ZF, or ZFC with the axiom of choice added), developed roughly 1908–1922 — doesn’t patch the definition of “set” so much as restrict which collections are allowed to count as sets in the first place, blocking Russell-style self-reference by construction. This is real machinery, and this site does not develop it: for everything below linear algebra, treating “set” at the intuitive level above is safe, and the axioms only start to matter if you push into genuine foundations of mathematics. That gap — full formal rigor exists, and almost nobody needs it day to day — is itself a recurring pattern worth noticing early.
4.3 Operations
A handful of operations on sets recur constantly enough to be worth having by name and symbol:
| Operation | Notation | Meaning |
|---|---|---|
| Union | \(A \cup B\) | elements in \(A\), or \(B\), or both |
| Intersection | \(A \cap B\) | elements in both \(A\) and \(B\) |
| Difference | \(A \setminus B\) | elements in \(A\) but not \(B\) |
| Complement | \(A^c\) | elements not in \(A\) (relative to some understood universe) |
| Power set | \(\mathcal{P}(A)\) | the set of all subsets of \(A\), including \(\varnothing\) and \(A\) itself |
| Cartesian product | \(A \times B\) | the set of ordered pairs \((a, b)\) with \(a \in A\), \(b \in B\) |
The Cartesian product deserves a second look, because it’s the one that stops being “obviously about sets” and starts being load-bearing for everything that follows: an ordered pair \((a, b)\) is not itself a primitive notion, it’s defined purely in terms of sets (one common construction is \((a,b) = \{\{a\}, \{a,b\}\}\)), which is precisely how a relation — and from it, a function — gets built out of nothing but sets in the next chapter.
4.4 Where this leads
Sets alone don’t do much; they need structure imposed on them before anything interesting happens. Relations and Functions is the first layer of that structure — the Cartesian product from this chapter turns out to be exactly the raw material a relation is built from.