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 $$A \times B,$$ which is the set of all possible ordered pairs $$\left( {a,b} \right),$$ where $$a \in A$$ and $$b \in B.$$

If $$R \subseteq A \times B$$ is a binary relation and $$\left( {a,b} \right) \in R,$$ 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 $$f \subseteq A \times B$$ that satisfies the following two properties:

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

As a counterexample, consider a relation $$R$$ that contains pairs $$\left( {1,1} \right),\left( {1,2} \right).$$ 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.