4. Functions and relations

Functions and relations

Functions are special binary relations.

A function $f:A\to B$ can be considered as a relation from $A$ to $B$:

$$\{(a,b)\in A\times B|f(a)=b\}$$

Relations that are also functions have two special properties, following from function rules:

• For every element $a\in A$, there is some $b\in A$ with $(a,b)$ being in the relation.
• No two ordered pairs in the relation have the same first element.

For example:

• Not a function: $\{(x,y)\in \mathbb{N}\times \mathbb{N}|y=x-1\}$ $0$ is not mapped.
• Function: $\{(x,y)\in \mathbb{Z}\times \mathbb{Z}|y=x-1\}$ $f(n)=n-1$
• Not a function: $\{(x,y)\in \mathbb{N}\times \mathbb{R}|x=y^2\}$ $(25,5)$ and $(25,-5)$ are both present in the relation.
• Not a function: $\{(x,y)\in \mathbb{N}\times \mathbb{N}|x=y^2\}$ $2$ is not mapped.
• Function: $\{(x,y)\in \mathbb{Z}\times \mathbb{Z}|y=x^2\}$ $f(n)=n^2$