3 Function and relation

Are functions and relations related? The answer is yes. In fact, a function is a binary relation with some constraints satisfied. One way to look at a function is as follows.

A function is a relation $L \subseteq X \times Y$ such that $\exists (i, j), (i, k) \in L \Rightarrow (j = k)$ . In this notation, $X$ is called the domain, and $Y$ is called the range. In plain English, it means that an element in the domain is related to at most one value in the range.

As an example, let us consider $X = \{a, b, c\}$ , and $Y = \{x, y\}$ . $L = \{(a, x), (b, x), (c, y)\}$ is a function. However, $L = \{(a, x), (a, y), (b, x), (c, y) \}$ is not a function.

In some definitions of a function, it is also required that $\forall e \in X: \exists (e, f) \in L$ . This means that every elements in $X$ maps to a value in $Y$ . Our example above satisfies this requirement. We will use this definitions (with the extra requirement) in this module.