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.


Reflexivity:


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


a+b = b+a


which is true since addition is commutative on N.


R is reflexive.


Symmetricity:


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.


Transitivity:


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


Hence,


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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