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Relations And Functions

Question
CBSEENMA12035649
Wired Faculty App

Let A= R × R and * be a binary operation on A defined by

(a, b) * (c, d) = (a+c, b+d)

Show that * is commutative and associative. Find the identity element for *

on A. Also find the inverse of every element (a, b) ∈ A.

Solution

(a, b) * (c, d) = (a + c, b + d)
(i) Commutative
(a, b) * (c, d) = (a+c, b+d)
(c, d) * (a, b) = (c+a, d+b)
for all, a, b, c, d ∈ R
* is commulative on A
(ii) Associative : ______
(a, b), (c, d), (e, f) ∈A
{ (a, b) * (c, d) } * (e, f)
= (a + c, b+d) * (e, f)
= ((a + c) + e, (b + d) + f)
= (a + (c + e), b + (d + f))
= (a*b) * ( c+d, d+f)
= (a*b) {(c, d) * (e, f)}
is associative on A
Let (x, y) be the identity element in A.
then,
(a, b) * (x, y) = (a, b) for all (a,b) ∈ A
(a + x, b+y) = (a, b) for all (a, b) ∈ A
a + x = a, b + y = b for all (a, b) ∈ A
x = 0, y = 0
(0, 0) ∈ A
(0, 0) is the identity element in A.
Let (a, b) be an invertible element of A.
(a, b) * (c, d) = (0, 0) = (c, d) * (a, b)
(a+c, b+d) = (0, 0) = (c+a, d+b)

a + c = 0 b + d = 0
a = - c b = - d
c = - a d = - b
(a, b) is an invertible element of A, in such a case the inverse of (a, b) is (-a, -b).

Some More Questions From Relations and Functions Chapter

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

 Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a)    R = {(x, y) : x and y work at the same place}
(b)    R = {(x,y) : x and y live in the same locality}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b3} is refleive, symmetric or transitive.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’. 

If R and R’ are symmetric relations on a set A, then R ∩ R’ is also a sysmetric relation on A.

Show that the union of two symmetric relations on a set is again a symmetric relation on that set.

Let A = {1. 2. 3}. Then show that the number of relations containing (1,2) and (2. 3) which are reflexive and transitive but not symmetric is four.

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