If we define a relation R on the set N times N as a b R c d hArr a d b...........

Mathematics (NDA) Relations and Functions

If we define a relation R on the set N × N as (a, b) R (c, d) ⇔ a + d = b + c for all (a, b), (c, d) ∈ N × N, then the relation is -

symmetric only

symmetric and transitive only

equivalence relation

reflexive only

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Explanation:

(a, b) R (c, d) ⇒ a + d = b + c
Or ⇒ b + c = a + d
⇒ (b, c) R (a, d)
∴ R is symmetric. →(1)
(a, b) R (c, d) ⇔ a + b = b + c
a + a = a + a
⇒ (a, a) R (a, a)
∴ R is reflexive. →(2)
(a, b) R (c, d) and (c, d) R (e, f)
⇒ a + d = b + c and c + f = d + e
⇒ a + d + c + f = b + c + d + e
⇒ a + f = b + e
⇒ (a, b) R (e, f)
∴ R is transitive. →(3)
From (1), (2), and (3): R is an
equivalence relation.