Summary Allen short notes (maths)

CBSE

CHA PTER -1 : RELA TION S AND FUNC TION S
CARTESIAN PRODUCT OF TWO sacrs :
Given two non-empt y sets /\ nnd B. ·1he c11rtcs ifln product /\ / B 1<1 th e <ict of all ord ered
pairs of
the fo nn (a, b) where th e first entry crn ncs firnn 'lC I /\ & c,cco nd comc<i from <;Ct U.
A '< U {(u,b)j a E A, h E B}
Example : A = { l. 2, 3}and B - {p, q}
/\ '< 13 - {( l , p),( I , q), (2, p), (2, q) , (3, p) , (3 , q) }
Note :
(i) If either A or B is a null set, then A x B will also be empty c:,et, 1.e. A / B ~
(ii) If n(A) = m, n(B) = n, then n(A x B) = mn.
2. RELATIONS:
Let A and B be two sets, then a relation R from A to B is a subset of A Y B.
Thus, R is a relation from A to B ~ R ~ A x B.
Note:
(i) If (a, b) E R then bis the image of a under Rand a is the pre-image of b under R.
(ii) If n(A) = m, n(B) = n, then the total number of relations defined from set A to B are 2cm.
3. DOMAIN AND RANGE OF A RELATION:
Let R be a relation from a set A to a set B. Then the set of all first components or coordina
tes of
the ordered pairs belonging to R is called the domain of R, while the set of all
second
components or coordinates of the ordered pairs in R is called the range of R.
Thus,Domain (R) ={a: (a, b) ER}
And, Range (R) = {b: (a, b) ER}
It is evident from the definition that the domain of a relation from A to B is a subset of.-\
and its
range is a subset of B.
illustration 1: Let A= {l, 2, 3, 4} and B = {x, y, z} . Consider the subset R = {( l. x). ( l. y). r2.
z).
(3, x)} of A x B. ls R, a relation from A to B? If yes, find domain and range of R. Draw arrow
diagram of R.
Solution: Since every subset of A x B is a relation from A to B.
therefore , R is also a relation from A to B.
⇒ Domain of R = {I , 2, 3} ; Range of R = { x, y, z}

4. TYPES OF RELATIONS :
{i) Void or Empty Relation : A relation R on a set A is calkd void or empty relation, if no
element of set A is related to any element of set A.
Example: The relation Ron the set A = { l, 2, 3, 4, 5} defin~d by R = {(a. b) : a - b = 12}.
We observed that a - b -:1; 12 for any two elements of A.
(a, b) ~ R for any a, b e A.
⇒ R does not contain any element of A x A .
⇒ R is empty set.
⇒ R is the void or empty relation on A.

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1

, Mathematics . • A relation Ron a set A is called universal relation ·r 4LLl'-t
·1·) Universal Relation . ,·y clement of set A. , I each elerne
(1 . . I ted to eve nt
of set A 1s ,ea . R 11 the set == {I, 2, 3, 4, 5, 6} defin ed by T :::: {(a b)
. The relation o , E R : la - bi >
ExarnPIe · - 0} .
We observe that' A
a - bl ~ 0 for all a, b E AxA
I , E R fonill (a,b) E .
::::> (d, b) t· , ·t A is related to every clemcnl of set A
::::>
. It clelllCllt o sc
EdC
::::> R == A XA
R is un iversa l relation on set A. .
:::::> • • T ··vial Relation means either each element of set A . ,
• · I Relation • 11 is re1atcd to
(iii) Tnv,a A . element of set A is related to any element of set A every
element of set 01 no . . . . .
relation or umversal relat10n are tnv1 al relation .
Example : BotI1 emPty . . . .
. R I t'1011 . A relation IAon A 1s called the identity relati on if every el
(iv) Identity e a • ement of A i,
related to itself only. .
. If A = {I 2 3} then the relat10n IA = {(I , 1), (2, 2) (3 3)' 1·s th .d .
Examp Ie . , ' . ' _ ' ' f e I en tity
relation on set A. But, relat10n R1 = {(I , I), (2, 2)} and R2 - {(I , I), (2, 2), (3, 3), (1 , 3)} are
not identity relations on set A, because (3, 3) ~ R, and in R2 element I is related to
elements I and 3.
(v) Reflexive Relation : A relation Ron a set A is said to be reflexive if every element of A is
atleast related to itself.
Thus, R is reflective <=> (a, a) ER v[§ AJ
Example : In a set A = {I, 2, 3}, relation R1 = {(I , I), (2, 2), (1, 2), (3, 3)} is
reflexive because every elements of set A is related to itself under R1, while relation
R2 = {(l , 1), (2, 2), (1, 3)} is not reflexive because (3, 3) ~ R2.
Note : Total number of reflexive relation on set A is 2n(n- 1), where n is number of elements in
set A.
(vi) Symmetric Relation : A relation R on a set A is said to be a symmetric relation
iff (a, b) E R ⇒ (b, a E R for all a, b E A
i.e., a Rb ⇒ b R a for all a, b E A.
Example : In a set A = {I, 2, 3}, relation R 1 = {(I, I), (2, 2), (I , 1), (2, 1)} is symmetric
because (a, b) E R1 ⇒ (b, a) E R1, V a, b EA, while relation R2 = {(I , 1), (3, 3), (1 , 3)} is
not symmetric because (3, I) ~ R2.
Note : Total b . n( n+I)
. num er of symmetnc relation on set A is 2 2 where n is number of
elements m set A. '
(vii) T
ransitive Relation . L t A b ..
relation iff ( b) · e e any set. A relation R on set A is said to be a tran sitive
a, E R and (b c) E R ( )
i.e., a Rb and b R ' ⇒ a, c E R for all a, b, c E A
E e ⇒ a R c for all a b c E A
xample : In a set A = {I ' ' ·s
2
transitive because ( b) ' , 3}, relation R, = {(I , I), (2, 2), (I , 2), (2, 1)}_1
R2 = {(I, 3), (3, 2)} ~~ no E R1 ~n~ (b, c) E R1 ⇒ (a, c) E R1, V a, b, c E A, while relat10n
t transitive because (I 2) R
' ~ 2·
2