# Functions as Relations

## Definition of a Function

Recall that a binary relation R from set A to set B is defined as a subset of the Cartesian product A × B, which is the set of all possible ordered pairs (a, b), where a A and b B.

If R A × B is a binary relation and (a, b) R, we say that a is related to b by R. It is denoted as aRb.

A function, denoted by f, is a special type of binary relation. A function from set A to set B is a relation f A × B that satisfies the following two properties:

• Each element a A is mapped to some element b B.
• Each element a A is mapped to exactly one element b B.

As a counterexample, consider a relation R that contains pairs (1, 1), (1, 2). The relation R is not a function, because the element 1 is mapped to two elements, which violates the second requirement.

In the next example, the second relation (on the right) is also not a function since both conditions are not met. The input element 11 has no output value, and the element 3 has two values - 6 and 7.

If $$f$$ is a function from set $$A$$ to set $$B,$$ we write $$f : A \to B.$$ The fact that a function $$f$$ maps an element $$a \in A$$ to an element $$b \in B$$ is usually written as $$f\left( a \right) = b.$$

## Domain, Codomain, Range, Image, Preimage

We will introduce some more important notions. Consider a function $$f : A \to B.$$

The set $$A$$ is called the domain of the function $$f,$$ and the set $$B$$ is the codomain. The domain and codomain of $$f$$ are denoted, respectively, $$\text{dom}\left({f}\right)$$ and $$\text{codom}\left({f}\right)$$.

If $$f\left( a \right) = b,$$ the element $$b$$ is the image of $$a$$ under $$f.$$ Respectively, the element $$a$$ is the preimage of $$b$$ under $$f.$$ The element $$a$$ is also often called the argument or input of the function $$f,$$ and the element $$b$$ is called the value of the function $$f$$ or its output.

The set of all images of elements of $$A$$ is briefly referred to as the image of $$A.$$ It is also known as the range of the function $$f,$$ although this term may have different meanings. The range of $$f$$ is denoted $$\text{rng}\left({f}\right)$$. It follows from the definition that the range is a subset of the codomain.

See solved problems on Page 2.