Proving Relation Properties

Intro to Proofs · Axiom Academy

EXAMPLE Proving Relation Properties Verify that a relation is an equivalence relation by checking reflexivity, symmetry, and transitivity. Define a relation on the integers by . Prove that R is an equivalence relation — that is, show it is reflexive, symmetric, and transitive. To prove R is an equivalence relation, we must verify: Excellent work! You've proven that R is an equivalence relation. Here's what we used: Equivalence relations: a relation is an equivalence relation exactly when it is reflexive, symmetric, and transitive — all three must be checked. Reflexivity: show every element is related to itself. Here x - x = 0 , which is always even. Symmetry: show forces . We used y - x = -(x - y) , and the negative of an even number is even. Transitivity: show and force . Adding (x - y) + (y - z) = x - z cancels y and stays even. Even-number facts: 0 is even, the negative of an even number is even, and the sum of two even numbers is even. This relation is exactly " x and y have the same parity." The same three-property checklist proves any equivalence relation — verify each property independently and carefully.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.