Injective, Surjective, Bijective

Three Types of Functions

Functions can be classified by how inputs and outputs relate.

• Injective — no output is hit more than once
• Surjective — every output is hit at least once
• Bijective — both injective and surjective

Injective (One-to-One)

A function is injective if different inputs always give different outputs.

$f(a) = f(b) \implies a = b$

No two inputs share the same output.

Surjective (Onto)

A function is surjective if every element in the codomain is hit by some input.

$\text{Range} = \text{Codomain}$

Nothing in the codomain is left unused.

Bijective (One-to-One and Onto)

A function is bijective if it is both injective and surjective.

• Each output is hit exactly once
• Every output is hit

Perfect pairing — each input matches exactly one output, and vice versa.

Summary

Why It Matters

Bijective functions have inverses.

If $f: A \to B$ is bijective, you can reverse it: $f^{-1}: B \to A$

This works because:

• Injective means no ambiguity going backwards
• Surjective means every element in B came from somewhere