Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

40 0. Introduction and Preliminaries
function is the same as f defined above. It is absolutely not.
Even though the rule is the same, the domain and codomain
are different, so these are two different functions.
Example 0.4.2
Just because you can describe a rule in the same way you would
write a function, does not mean that the rule is a function. The
following are NOT functions.
1. f : N → N defined by f (n) n 2
. The reason this is not a
function is because not every input has an output. Where
does f send 3? The rule says that f (3) 3 2 , but 3 2
is not an
element of the codomain.
2. Consider the rule that matches each person to their phone
number. If you think of the set of people as the domain and
the set of phone numbers as the codomain, then this is not a
function, since some people have two phone numbers. Switching
the domain and codomain sets doesn’t help either, since
some phone numbers belong to multiple people (assuming
some households still have landlines when you are reading
this).
Describing Functions
It is worth making a distinction between a function and its description.
The function is the abstract mathematical object that in some way exists
whether or not anyone ever talks about it. But when we do want to talk
about the function, we need a way to describe it. A particular function can
be described in multiple ways.
Some calculus textbooks talk about the Rule of Four, that every function
can be described in four ways: algebraically (a formula), numerically (a
table), graphically, or in words. In discrete math, we can still use any of
these to describe functions, but we can also be more specific since we are
primarily concerned with functions that have N or a finite subset of N as
their domain.
Describing a function graphically usually means drawing the graph of
the function: plotting the points on the plane. We can do this, and might
get a graph like the following for a function f : {1, 2, 3} → {1, 2, 3}.