Q. 114.7( 3 Votes )

Let A = {1, 2, 3,

Answer :

Given that, A = {1, 2, 3, ... 9} and R be the relation in A ×A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A ×A.


Let (a,b) R (a,b)

a+b = b+a

which is true since addition is commutative on N.

R is reflexive.


Let (a,b) R (c,d)

a+d = b+c

b+c = a+d

[since addition is commutative on N]

c+b = d+a

(c,d) R (a,b)

R is symmetric.


Let (a,b) R (c,d) and (c,d) R (e,f)

a+d = b+c and c+f = d+e

(a+d) – (d+e) = (b+c ) – (c+f)

a-e= b-f

a+f = b+e

(a,b) R (e,f)

R is transitive.

Hence, R is an equivalence relation.

Equivalence class [(2,5)]

Let (a,b) R (2,5)

a+5 = b+2

a-b = 2-5

= -3


The equivalence class [(2,5)] = {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}

In general,

{(a,a+3)|a A}

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