Relations and Functions Summary

Set Theory · Axiom Academy

SUMMARY Summary: Relations and Functions Let's review the key concepts from Unit 3: Relations and Functions Ordered Pairs: Unlike sets, order matters in (a,b) . Two ordered pairs are equal iff both components match. Cartesian Product: A × B contains all ordered pairs with first element from A , second from B . Relations: A relation is any subset of A × B , connecting elements from two sets. Why It Matters: Relations generalize the notion of "relationship" between mathematical objects. Reflexive: Every element relates to itself ( aRa ) Antisymmetric: If aRb and bRa , then a = b Transitive: If aRb and bRc , then aRc Example: Modular Arithmetic as Equivalence Relation Define the Relation: On integers ℤ , say aRb if a ≡ b (mod 3) Check Properties: Reflexive: a ≡ a (mod 3) ✓ Symmetric: If a ≡ b (mod 3) , then b ≡ a (mod 3) ✓ Transitive: If a ≡ b and b ≡ c (mod 3) , then a ≡ c (mod 3) ✓ Find Equivalence Classes: [0] = ..., -6, -3, 0, 3, 6, ... , [1] = ..., -5, -2, 1, 4, 7, ... , [2] = ..., -4, -1, 2, 5, 8, ... Result: The equivalence classes partition ℤ into three disjoint sets Definition: A function f: A → B is a relation where each input has exactly one output Domain: The set A of all inputs Codomain: The set B where outputs live Range: The subset of B that is actually used as output Injective (1-1): Different inputs give different outputs Surjective (onto): Every element in codomain is hit Bijective: Both injective and surjective (perfect pairing)

This is the written version of the interactive lesson above. See the full Set Theory course.