Injective and Surjective Functions

Set Theory · Axiom Academy

LESSON Injective and Surjective Functions Two fundamental properties: one-to-one and onto 1. Injective Functions (One-to-One) A function f: A → B is injective (or one-to-one ) if different inputs always produce different outputs: Definition: f is injective if f(a₁) = f(a₂) implies a₁ = a₂ Equivalently: If a₁ ≠ a₂, then f(a₁) ≠ f(a₂) Intuition: No two different inputs can map to the same output. Each output has at most one "pre-image." Example: f: ℝ → ℝ defined by f(x) = 2x is injective (doubling never maps two different numbers to the same result) Non-example: g(x) = x² on ℝ is NOT injective since g(-2) = g(2) = 4 2. Surjective Functions (Onto) A function f: A → B is surjective (or onto ) if every element of the codomain is hit by some input: Definition: f is surjective if for all b ∈ B, there exists a ∈ A such that f(a) = b Equivalently: Range(f) = B (the range equals the entire codomain) Intuition: The function "covers" the entire codomain—no element of B is left out. Example: f: ℝ → ℝ defined by f(x) = x³ is surjective (every real number has a cube root) Non-example: g: ℕ → ℕ defined by g(n) = 2n is NOT surjective (odd numbers are never hit) 3. Visual Understanding with Mapping Diagrams Mapping diagrams make these concepts crystal clear: Injective: No element in B has more than one arrow pointing to it Surjective: Every element in B has at least one arrow pointing to it Neither: Some element in B has multiple arrows, or some have none

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