Injective & Surjective Functions

One-to-one, onto, bijective — and counting injections.

The idea

A function is injective when no two inputs share an output and surjective when no element of the codomain is missed; together, the two properties are exactly what a function needs in order to be invertible.

A function $f \colon A \to B$ assigns to each element of $A$ exactly one element of $B$. The definition leaves two independent questions open: whether two different inputs may receive the same output, and whether every element of $B$ is an output at all. $f$ is injective, or one-to-one, when distinct inputs have distinct outputs — equivalently, when $f(a_1) = f(a_2)$ forces $a_1 = a_2$. $f$ is surjective, or onto, when every element of $B$ equals $f(a)$ for at least one $a$. A function with both properties is a bijection.

A bijection is exactly an invertible function. An inverse must send each $b$ in $B$ back to the input it came from, so such an input must exist — surjectivity — and there must be only one — injectivity. Therefore $f$ has an inverse if and only if it is a bijection.

Ways to work on it

• Walkthrough. Injective, surjective, bijective on familiar functions.
• Practice. Classify a function between finite sets.
• Hardest. Count the injections between finite sets, and decide when none can exist.

Not sure where to start? Take the ten-question placement test.