Sets and Relations Summary

Intro to Proofs · Axiom Academy

How sets form the foundation of all mathematics, and how relations and functions connect elements to model structure and computation. Foundation of mathematics: virtually every mathematical object — numbers, functions, structures — can be constructed from sets. Proof by definition: most set and relation proofs follow directly from carefully applying the definitions with logical precision. Relations generalize functions: a function is a relation with the special property that each input determines exactly one output. Equivalence relations classify: they give a rigorous way to group objects that are the same in some respect. Bijections preserve structure: a bijection establishes when two sets have the same structure, letting us transfer problems between domains. Core Concept Set Operations & Identities Union, intersection, complement, and difference combine sets in fundamental ways. Identities like De Morgan's laws and distributivity let you rewrite set expressions algebraically. Element-chasing: prove A = B by showing . Distributive law: — mirrors multiplication over addition. Watch out for: a Venn diagram gives intuition, not a rigorous proof. Core Concept Properties of Relations A relation R is described by which structural properties it satisfies. These four properties are the vocabulary for classifying every relation. Reflexive: — e.g. equality, divides on positive integers. Symmetric: — e.g. is a sibling of . Transitive: — e.g. ancestor of , .

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