Relations and Functions — Formula Sheet

Empty Relation

R = φ (no element is related to any other element). Example: R = {(a,b): a – b = 10} in A = {1,2,3,4} is empty.

Universal Relation

R = A × A (every element is related to every element). Example: R = {(a,b): |a – b| ≥ 0} in A = {1,2,3,4} is universal.

One-One (Injective) Function

Different inputs always give different outputs. f(x₁) = f(x₂) ⟹ x₁ = x₂.

Many-One Function

Two or more different inputs can give the same output. Example: f(x) = x² is many-one since f(1) = f(–1) = 1.