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Question

Let R be a relation over the set N×n and it is defined by (a, b) R (c, d) a+ d = b + c. Then, R is

A
reflexive only
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B
symmetric only
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C
transitive only
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D
an equivalence relation
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Solution

The correct option is D an equivalence relation
(a, b) R (a, b) because a + b = b + a. So, r is reflexive.
(a, b)R (c, d) a+d = b+c c+b = d+a
(c,d) R (a,b)
So, R is symmetric.
(a, b) R (c, d) and (c, d) R (e, f)
a + d = b + c, c + f = d + e
Adding, a + d + c + f = b + c + d +e
a + f = b + e
(a, b) R (e, f).
R is transitive.

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