Properties of Functions

Intro to Proofs · Axiom Academy

LESSON Properties of Functions Functions as special relations — and the three ways a mapping can behave: injective, surjective, and bijective. 1. Functions as Special Relations A function f from set A to set B (written ) is a relation in which every element of A (the domain ) is assigned to exactly one element of B (the codomain ). A function is injective (one-to-one) if different inputs always produce different outputs. No two elements of the domain land on the same element of the codomain. Assume f(x_1) = f(x_2) for arbitrary . Use algebra to deduce x_1 = x_2 . A function is surjective (onto) if every element of the codomain is the output of at least one element of the domain. The function "covers" the entire codomain. Construct an element — this is the key step. Since b was arbitrary, conclude that f is surjective. 4. Bijection (One-to-One Correspondence) A function is bijective if it is both injective and surjective. Every codomain element is mapped to by exactly one domain element — a perfect pairing between the two sets. f is injective (one-to-one), and • Perfect pairing between the sets • Every output has exactly one input • Every input has a unique output • Bijections have inverses • The two sets share the same cardinality • Establishes a one-to-one correspondence Surjectivity: for any , construct with f(a) = b . You've seen how a function is a special relation, and the three ways a mapping can behave — captured entirely by where its arrows land.

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