MATH1231 - Functions

Injective / One-to-One

A function is injective if no point in the codomain of the function is the function value of more than one point in the domain.

Essentially each y corresponds to only one x.

(1)
\begin{align} f : X \rightarrow Y \text { is injective if } f(x_1) = f(x_2) \text { only when } x_1 = x_2 \end{align}

Surjective / Onto

A function is surjective if every point in the range of the function is in the range. I.e. range = codomain.

Essentially every y exists.

(2)
\begin{align} f : X \rightarrow Y \text { is surjective if } \forall y \in Y \exists x \in X \text { such that } y = f(x) \end{align}

Bijective

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

Essentially every x has one y, and every y has an x.

Inverse

A function has an inverse iff the function is bijective.

A function is an inverse if:

  1. g o f = idX (i.e. idX(x) = x)
  2. f o g = idY (i.e. idY(y) = y)