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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)].

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one month ago Anand Kumar Pandey
1583 Points
```							Dear StudentGiven thatA = {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 + ea + d = b + c and d + e = c + f(a + d)–(d + e = (b + c)–(c + f) a–e=b–fa+f=b+e(a, b) R(e, f)So, R is transitive.Therefore, R is an equivalence relation.Thanks
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one month ago
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