Let `A=NxNa n d^(prime)*'` be a binaryoperation on A defined by `(a , b)*(C , d)=(a c , b d)` for all `a , b , c , d , in Ndot` Show that `'*'` is commutative and associative binary operation on A.

Explanation

`A=N times N` and` (a, b) *(c, d)=(a d+b c, b d)`

(i) Let` (a, b),(c, d),(e, f) `be any three elements of` A.`

Therefore,`{({a}, {b}) *(c, {~d})} *({e}, {f})=({ad}+{bc}, {bd}) *({e}, {f})`
`
=((a d+b c) f+(b d) e (bd) f)`

Therefore,`{(a, b) **(c, d)} *(e, f)=(a d f+b t f+b d e, b d f) quad ldots(1)`

Again `(a, b) *{(c, d) *(e, f)}=(a, b) *(c f+d e, d f)`
`
=(a(d f)+b(c f+d e), b(d f))`

Therefore, `(a, b) *{(c, d) *(e, f)}=( adf +b e f+b d e, b d f) quad ldots(2)`

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