Functions as a Special Kind of Relation; Pictorial Representation

Function. A relation $f$ from a non-empty set $A$ to a non-empty set $B$ is called a function (or mapping) from $A$ to $B$ if every element of $A$ appears as the first component of exactly one ordered pair of $f$ — that is, for every $a \in A$ there is one and only one $b \in B$ such that $(a, b) \in f$. This unique $b$ is written $f(a)$ and called the image of $a$ under $f$ (or the value of $f$ at $a$); $a$ is called the pre-image of $b$. A function from $A$ to $B$ is written $f : A \to B$.

So a function is a relation with two extra conditions layered on top of "any subset of $A \times B$": (i) every element of $A$ must be used as a first component (nothing in $A$ is left unrelated — this is what makes the domain equal to the whole of $A$, not merely a subset of it), and (ii) no element of $A$ may be used as the first component of two different ordered pairs (this is what makes the image unique). A relation that fails either condition is a relation but not a function.

Pictorial representation. In an arrow diagram for a function $f : A \to B$, these two conditions become a simple visual rule: exactly one arrow must leave every point of $A$. Elements of $B$, on the other hand, may receive no arrows, one arrow, or several arrows — there is no restriction on $B$'s side. This single-arrow-out-of-every-$A$-point picture is the quickest way to check, from a diagram, whether a given correspondence is a function.

Domain, co-domain and range of a function. For $f : A \to B$:

• The domain of $f$ is $A$ itself (by the definition above, every element of $A$ is used, so the domain is never a proper subset of $A$ the way it can be for a general relation).
• The co-domain of $f$ is $B$, again fixed as part of specifying $f$.
• The range of $f$ is the set of images actually taken: $\text{Range}(f) = \{f(a) : a \in A\}$, a subset of the co-domain $B$ that need not be all of $B$. …