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Let A = N xx N and * be the binary opera...

Let `A = N xx N` and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative.

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`(a, b) ** (c, d) = (a+c, b, d)`
For commutative `(a, b) ** (c, d) = (c, d) ** (a, b)`
`(c, d) ** (a, b) = (c+a, d + b)`
Therefore A is commutative
`[(a, b) ** (c, d)] ** (e, f) = (a + c, b+d) ** (e, f) = (a + c + e, b+ d + f)`
`(a, b) ** [(c, d) ** (e, f)] = (a, b) ** (c + e, d +f) = (a + c + e, b + d + f)`
Hence A is associative
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