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Prove that the relation R on the set Nxx...

Prove that the relation `R` on the set `NxxN` defined by `(a ,\ b)R\ <=>(c ,\ d) a+d=b+c` for all `(a ,\ b),\ (c ,\ d) in NxxN` is an equivalence relation. Also, find the equivalence classes [(2, 3)] and [(1, 3)].

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`(a, b) R(c, d) \Leftrightarrow a+d=b+c`

(i) `a+a=a+a`.

`\therefore(a, a) R(a, a)` implies `R` is reflexive.

(ii) `(a, b) R(c, d) \Rightarrow a+d=b+c`

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