Functions, Domain, and Range

Relations as functions

We can view functions as a special kind of relations because every function from $A$ to $B$ is a special subset of $A\times B$. A function $f$ from the set $A$ to the set $B$ is a subset of $A\times B$ that contains exactly one ordered pair $(a,b)$ for each $a\in A$ for some $b\in B$. We can write such a function in a compact way as $f: A\to B$. This means that a function from $A$ to $B$ relates each element of $A$ to exactly one element of $B$. However, an element of $B$ can be related to more than one element of $A$ or even none.

Note: Remember that every function is also a relation, but a relation may or may not be a function.

Terminology

A function from $A$ to $B$ is also called a map or a mapping from $A$ to $B$ and can also be referred as a function$\text{from }A\text{ into } B$.

If a function $f: A\to B$ contains an ordered pair $(a,b)$, we write it as $f(a)=b$. In this function, we can say that the element $b\in B$ is the image of the element $a\in A$ under the function $f$. Also, the element $a\in A$ can be called a preimage of $b\in B$ under $f$.

Examples

Let’s take the following two sets: