Set
Calculus & Advanced MathA set is a well-defined collection of distinct objects called elements, and is one of the most fundamental building blocks of mathematics.
Definition
A set is a collection of things grouped together. The things inside are called elements or members. A set must be well-defined: for any object, you can tell whether it belongs or not.
Example
The set of vowels in English: $\{a, e, i, o, u\}$. Is "b" in the set? No. Is "e"? Yes. The curly braces $\{\ \}$ are the standard notation for sets.
Key Insight
Sets are the basic language of all of mathematics. Almost every mathematical object, from numbers to functions, can be defined using sets.
Definition
A set $S$ is a collection of distinct, unordered objects. Membership is denoted $x \in S$. Key examples: $\mathbb{N} = \{1, 2, 3, \ldots\}$ (natural numbers), $\mathbb{Z} = \{\ldots, -1, 0, 1, \ldots\}$ (integers), $\mathbb{Q}$ (rationals), $\mathbb{R}$ (reals). Two sets are equal if and only if they have exactly the same elements.
Example
$\{1, 2, 3\} = \{3, 1, 2\}$ (order does not matter). $\{1, 1, 2\} = \{1, 2\}$ (duplicates are collapsed). A set can be described by listing elements or by set-builder notation: $\{x \in \mathbb{R} : x > 0\}$.
Key Insight
The Axiom of Extensionality formalizes equality: two sets are equal if every element of one is an element of the other and vice versa.
Definition
In Zermelo-Fraenkel set theory (ZFC), sets are defined axiomatically. Key axioms include Extensionality, Pairing, Union, Power Set, and the Axiom of Choice. Russell's paradox (the set of all sets not containing themselves) showed naive set theory is inconsistent and motivated axiomatic foundations.
Example
Russell's paradox: let $R = \{x : x \notin x\}$. Is $R \in R$? If yes, then $R \notin R$ (contradiction). If no, then $R \in R$ (contradiction). ZFC avoids this by restricting comprehension.
Key Insight
ZFC set theory is the standard foundation for modern mathematics. Independence results (like the Continuum Hypothesis being independent of ZFC) show there are mathematical questions ZFC cannot settle.