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Statement-1: The relation R on the set N...

Statement-1: The relation R on the set `N xx N` defined by (a, b) R (c, d) `iff` a+d = b+c for all a, b, c, d `in` N is an equivalence relation.
Statement-2: The intersection of two equivalence relations on a set A is an equivalence relation.

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(i) (a,b) R (a,b) implies a + b = b + a
`therefore` R is reflexive.
(ii) (a, b) R (c, d) implies a + d = b + c
`implies c+b=d+aimplies(c,d)R(a,b)`
`therefore R` is symmetric.
(iii) (a, b) R (c, d) and (c, d) R (e, f) implies a + d = b + c and c + f = d + e
`impliesa+d+c+f=b+c+d+e`
`impliesa+f=b+eimplies(a,b)R(e,f)`
`therefore R` is transitive.
Thus, R is an equivalence relation on `NxxN`.
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