Combinatorics Through Guided Discovery, 2004a

A.1. Relations as sets of Ordered Pairs 137
(f) Use the digraph of the previous part to explain whether the function
is one-to-one.
(g) Suppose the function f has domain S ′ and range T. Draw the digraph
of f and use it to explain whether f is onto.
(h) Use the digraph of the previous part to explain whether f is one-toone.
A one-to-one function from a set X onto a set Y is frequently called a bijection,
especially in combinatorics. Your work in Problem 337 should show you that a
digraph is the digraph of a bijection from X to Y
• if the vertices of the digraph represent the elements of X and Y,
• if each vertex representing an element of X has one and only one arrow
leaving it, and
• each vertex representing an element of Y has one and only one arrow entering
it.
Problem 338. If we reverse all the arrows in the digraph of a bijection f ,we
get the digraph of another function g. Is g a bijection? What is f (g(x))?
What is g( f (x))?
If f is a function from S to T, ifg is a function from T to S, and if f (g(x)) = x for
each x in T and g( f (x)) = x for each x in S, then we say that g is an inverse of f
(and f is an inverse of g).
More generally, if f is a function from a set R to a set S, and g is a function from
S to T, then we define a new function f ◦ g, called the composition of f and g ,by
f ◦ g(x) = f (g(x)). Composition of functions is a particularly important operatio
in subjects such as calculus, where we represent a function like h(x) = √ x 2 +1as
the composition of the square root function and the square and add one function
in order to use the chain rule to take the derivative of h.
The function ι (the Greek letter iota is pronounced eye-oh-ta) from a set S to
itself, given by the rule ι(x) =x for every x in S, is called the identity function on
S. If f is a function from S to T and g is a function from T to S such that g( f (x)) = x
for every x in S, we can express this by saying that g ◦ f = ι, where ι is the identity
function of S. Saying that f (g(x)) = x is the same as saying that f ◦ g = ι, where
ι stands for the identity function on T. We use the same letter for the identity
function on two different sets when we can use context to tell us on which set the
identity function is being defined.
Problem 339. If f is a function from S to T and g is a function from T to S
such that g( f (x)) = x, how can we tell from context that g ◦ f is the identity
function on S and not the identity function on T? (h)