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Let 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. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].

 Let 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.
Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]. 

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
2 years ago
Dear Student
Given that
A = {1, 2, 3, ... 9} and (a, b) R (c, d) if a + d = b + c for (a, b), (c, d)∈A ×A.
Let (a, b) R(a, b)
So, a + b = b + a,∀a, b∈A which is true for any a, b∈A.
Thus, R is reflexive.

Let (a, b) R(c, d) Then, a+d=b+c c+b=d+a

(c, d) R(a, b)
Thus, 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 = b + c and d + e = c + f
(a + d)–(d + e = (b + c)–(c + f) a–e=b–f
a+f=b+e
(a, b) R(e, f)
So, R is transitive.
Therefore, R is an equivalence relation.

Thanks

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