# Functions and Inverse Functions

Asterisks denote definitions which are **not **in the
textbook.

1. We review the definitions leading up to the textbook's definition of inverse
function.

(a) *A relation is a set of ordered pairs .

(b) If there are sets R, A, and B such that

R ⊆ A × B,

then R is said to be a relation between A and B.

(Clearly, if R is a relation between A and B, then R is a relation as defined in
(a).)

(c) *The domain of a relation R, denoted by dom(R), is defined by

(d) *A relation R is called a function, or is said to have the function
property , iff

∀ a, b, c, and d, if (a, b) ∈ R and (c, d) ∈ R and a = c, then b = d.

In other words, a relation R is a function iff

if (a, b) ∈ R and (a, d) ∈ R then b = d.

(e) Given sets A and B, a relation F is said to be a** function from A to B**
iff

F is a relation between A and B;

F has the function property ;

dom(F ) = A (note that the fact that F is a relation between A and B implies
that ran(F) ⊆ B ).

The notation F : A → B means that F is a function from A to B, although the
exact verbal

interpretation of the symbols F : A → B may differ slightly depending on the
context.

(f) Theorem. *For a function F : A → B, define the relation F^{-1} by

**(I) **(b, a) ∈ F^{-1} iff (a, b) ∈ F , (I)

(i.e.,

z ∈ F^{-1} iff ∃ a, b z = (b, a) and (a, b) ∈
F )

If the function F : A → B is** bijective, **then F ^{-1} is a
function from B to A. (You should be

able to prove this.)

(g) The definition of inverse function given in the textbook can be expressed in
the following way:

If a function F : A → B is bijective, then the function F^{-1}: B → A
defined above is called the

**inverse function** of F : A → B.

*To distinguish this concept from another inverse function concept discussed
below, we will

temporarily refer to this as the L-**inverse function** of F : A → B.

(The letter “L” refers to the author of the textbook.)

*I.e., a function G : B → A is the L-inverse function of a bijective function F
: A → B iff

dom(G) = B and G is the inverse relation F^{-1} of the relation F as defined in
(I) above.

(h) *Expressed in standard function notation instead of with the use of ordered
pairs , equation (I)

above gives the following defining condition for the L-inverse:

For every a in A and b in B ,

**(I')** F^{-1}(b) = a iff F(a) = b.

We will give another definition of “inverse function”, and
then we will comment on the definitions and

the relationships between them.

2. *Given a function f : A → B, **a** function g : B → A is called an** inverse**
of the function f : A → B

iff

∀ x in A, g (f (x)) = x,

and

∀ y in B, f (g (y)) = y.

We will temporarily refer to such a function g : B → A as an **
algebraic -inverse function** of

f : A → B .

It should be clear that this definition is equivalent to the following :

For any set S, let id_{S} denote the identity function on S.

Given a function f : A → B, a function g : B → A is called an **
( algebraic -)inverse** of the

function f : A → B iff

g o f = id_{A}

and

f o g = id_{B} .

**Comments:
**

**Uniqueness.**It is clear that the L-inverse of a function is unique, if it exists, since the L-inverse

function is obtained in a particular way from the original function. On the other hand, if the inverse

function is defined as an algebraic -inverse, then it has to be proved that if such an inverse function

exists, then it is unique.

**Equivalence of L- inverse and algebraic -inverse.**You should be able to prove (this is related to

Ex. 7.29 in the textbook) that the L-inverse idea and the algebraic -inverse idea are equivalent in the

following sense:

A function f : A → B has an algebraic -inverse function iff f : A → B is bijective.

A function g : B → A is the L-inverse function of f : A → B iff g : B → A is an algebraic -inverse

function of f : A → B.

**Note that this implies that if an algebraic-inverse function exists, then it is unique.**

Uniqueness.

Uniqueness.

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