Surjection (Onto Function)

Definition

A function \(f: A \to B\) is a surjection (or onto) if:

Equivalently: every element in the codomain is the image of some element in the domain.

Examples

Example 1: \(f(x) = x^3\) on \(\mathbb{R}\)

For any \(y \in \mathbb{R}\), take \(x = \sqrt[3]{y}\). Then \(f(x) = y\).

This is a surjection.

Example 2: \(f(x) = x^2\) on \(\mathbb{R} \to \mathbb{R}\)

There is no \(x\) such that \(f(x) = -1\).

Not a surjection (but is a surjection onto \([0, \infty)\)).

Example 3: Projection

\(\pi_1: A \times B \to A\) given by \(\pi_1(a, b) = a\) is a surjection.

Composition of Surjections

If \(f: A \to B\) and \(g: B \to C\) are surjections, then \(g \circ f: A \to C\) is a surjection.

Related

• Function — The general concept
• Injection — Distinct inputs map to distinct outputs (dual concept)
• Bijection — Both injection and surjection
• Cartesian Product — Used in projection example
• Direct Proof — Proving a function is surjective