RELATIONS and FUNCTIONS

RELATIONS

1. INTRODUCTION

In class XI, we have introduced the notion of a relation, its

domain, co-domain and range. Let us recall that a relation form

a set A to a set B is a subset of A X B. if R is a relation from a set

A to a set B and (a,b) Є R, then we say that a is related to b

under relation R and we write as a R b.

1. A relation form a set A to a set B is a subset of A X B.

2. Total number of relations from a set consisting of m

elements to a set consisting of n element is 2mn.

3. A relation on a set A is said to be A X A.

4. A relation R on set A is said to be

i) The identity relation, if every element of A is related to

itself only.

ii) Reflexive, if (a, a) Є R for all a Є A.

iii) Symmetric, if (a, b) Є R⇒(b, a) Є R for all a, b Є A.

iv) Transitive, if (a, b) Є R and (b,c) Є R ⇒ (a, c) Є R for all a,

b, c Є A.

v) An equivalence relation, if it is reflexive, symmetric and

transitive.

vi) Antisymmetric, if (a, b) Є R and (b, a) Є R ⇒ a = b

vii) The empty relation, if R = ф

viii) The universal ration, if R = A X A.

FUNCTIONS

1. Let A and B be two non-empty sets. Then, a subset f of

A X B is a function from A to B, if

i) For each a Є A there exists b Є B such that (a, b) Є f

ii) (a, b) Є f and (a,c) Є f ⇒ b = c.

In other words, a subset f of A X B is a function from A to

B, if each element of A appears in some ordered pair in f

and no two ordered pairs in f have the same first

element.

2. Let A and B be two non-empty sets. Then, a function f from

A to B associates every element of A to a unique element of

B. the set A is called the domain of f and the set B is known

as its co-domain. The set of images of elements of set A is

known as the range of f.

3. If f: A B is a function, then x = y ⇒ f (x) = f (y) for all x, y

Є A.

4. A function f : A B is a one-one function or an injection, if

F(x) = f(y) ⇒ x = y for all x, y Є A or, x ≠ y ⇒ f (x) ≠ f(y) for

all x, y Є A

Graphically, if the graph of a function does not take a turn,

in other words a straight line parallel to x-axis does not cut the

curve at more than one point, and then it is a one-one function.

Note that a function is one-one, if it is either strictly increasing

or strictly decreasing.

5. A function f: A B is an on to function or a surjection, if

range (f) = co-domain (f).

6. Let A and B be two finite sets and f:A B be a function.

7. If A and B are two non-empty finite sets containing m and

n elements respectively, then

i) Number of functions from A to B = nm.

ii) Number of one-one functions from A to B =

{ nCm x m!, if, n ≥ m }

0, if n < m

iii) Number of on to functions from A to B =

∑ (−1)ᶯˉʳ𝑛

𝑟=1 nCr rm, if m ≥ n

iv) Number of one-one and onto functions from A to B =

n!, if m = n

0, if m ≠ n

8. If a function f : A B is not an on to function, then

f : A f (A) is always an on to function.

9. The composition of two bisection.

10. If f: A B is a bisection, then g: B A is inverse of f, iff

f (x) = y ⇒ g(y) = x

Or, gof = IA and fog = IB

11. Let f : A B and g:B A be two functions.

i) If gof = IA and f is an injection, then g is a surjection.

ii) If fog = IB and f is a surjection, then g is an injection.

12. Let f : A B and g : B C be two function. Then

i) gof : A C is onto ⇒ g :B C is onto.

ii) gof : A C is one-one ⇒ f:A B is one-one.

iii) gof : A C is onto and g : B C is one-one ⇒ f ; A B is

onto.

iv) gof : A C is one-one and f:A B is onto ⇒ g : B C is

one –one.