Functions and Relations 129

A function f : A → B is bijective if it is both injective and surjective, or in other words,

if it is both one-to-one and onto. In this case, we say that f puts the sets A and B into

a one-to-one correspondence.Iff : A → B and g : B → C, then we may define the

composition function g ◦ f as a function from A to C by specifying how it acts on arbitrary

input: (g ◦ f )(a) = g( f (a)). This is indeed a function from A to C, since every a ∈ A

determines the value of f (a), which determines the value of g( f (a)), and so for every a ∈ A

there is a unique value for the composition (g ◦ f )(a).

f

g

A

B

C

a

f (a)

g( f (a))

= (g ◦ f )(a)

g ◦ f

Theorem 94. The composition of injective functions is injective.

Proof. Suppose that f : A → B and g : B → C are both injective. Consider the compo-

sition function h = g ◦ f , which is a function from A to C. Suppose that a  a



are two

distinct points in A. Since f is injective, it follows that f (a)  f (a



) are distinct points in B.

Since g is injective, it follows that g( f (a))  g( f (a



)), and consequently that h(a)  h(a



).

So h is injective, as desired.

If f : A → B is an injective function, then we may define the inverse function f

−1

, whose

domain will be the range of f .Ifb is in the range of f , then b = f (a) for some a ∈ A, and

we define f

−1

(b) = a. This is a function, since the injectivity of f means that a is unique,

given b , since no other a



 a can have f (a



) = f (a). Thus, the object b in the range of

f determines the object a in A from which it arose, and so f

−1

(b) has only one possible

value a.

Mathematical Habits

Identify and name the key concepts. Identify the important concepts, and invent

precise terminology so that you may recognize and refer to that phenomenon when it

arises. It is difficult to reason about an idea when you have no meaningful way to refer

to it. So give yourself a way to refer to it by naming it with suitable terminology.