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Let A=NxN , and let * be a binary operat...

Let `A=NxN ,` and let * be a binary operation on A defined by `(a , b)*(c , d)=(a d+b c , b d)` for all `(a , b),c , d) in NxNdot` Show that : `'*'` is commutative on `A` `'*^(prime)` is associative on`A` `A` has no identity element.

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(i) Commutativity: `(a, b),(c, d) in N times N`

`(a, b) **(c, d)=(a+c, b+d)=(c+a, d+b)`

`(a, b, c, d in N, a+c=c+a b+d=d+c) `

` =(c, d) ** b`

So `, (a, b) **(c, d)=(c, d) **(a, b)`

` **` is commutative.

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