If we define a relation R on the set N x N as a b R c d a d b c for all a b c d N x N then the ...........

Mathematics (NDA) Relations and Functions

If we define a relation R on the set N x N as (a, b) R (c, d) $\Leftrightarrow$ a + d = b + c for all (a, b), (c, d) $\varepsilon$ N x 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) $\Leftrightarrow$ a + b = b + c
a + a = a + a
=> (a, a) R (a, a) => R is reflexive.
Next, Let (a, b) R (c, d) => a + b = b + c
=> c + b = d + a => (c, d) R (a, b)
=>R is symmetric.
Next, Let (a, b) R (c, d) and (c, d) R (e, f)
=> a + b = 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 => R is an equivalence relation.