The Basics of Functions

Definition. A function f from a set X to a set Y is a relation which satisfies the following two conditions:

  1. \forall x\in X, \exists y\in Y: (x,y)\in f, and
  2. \forall x\in X, y_1,y_2\in Y, \left[(x,y_1)\in f \wedge (x,y_2)\right]\in f\Rightarrow y_1=y_2

That is, a function assigns to each (clause 1) element of X an element of Y, and that assignment is unambiguous (clause 2). Since we could rewrite clause 1 as \operatorname{Domain}(f)=X (why?), we'll refer to clause 1 as the "domain clause" and clause 2 as the "unambiguousness clause".

Notation. Instead of the mouthful f is a function from X to Y, we write the shorthand f:X\rightarrow Y.

We also have this familiar notation:

Notation. If f is a function from X to Y, and (x,y)\in f, we write y=f(x).

Observe that if (x,y_1)\in f and (x,y_2)\in f, then this notation reads as follows: y_1=f(x) and y_2=f(x), so we'd darn well better have y_1=y_2, or else something is very wrong.

There is another way to represent a function, by specifying its domain and its rule:

Notation. If f is a function from X to Y, we write f:X\rightarrow Y. To specify the rule that f uses, we write

    \begin{align*}f:X&\rightarrow Y\\x&\mapsto f(x) \end{align*}

For example, we could denote the function which takes in a real number as input, and squares it by

    \begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ t&\mapsto t^2 \end{align*}

Notice that the arrow between the sets is a different shape from the one between elements of the set. As usual we want to let go of our xs, so we could as well have written

    \begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ z&\mapsto z^2 \end{align*}


    \begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ Q&\mapsto Q^2 \end{align*}

This frees us from the tyranny of always writing y=f(x).