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Let A=NxxN , and let * be a binary op...

Let `A=NxxN` , 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 NxxNdot` Show that `A` has no identity element.

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Identity element
`a*3=a=e*a`
Let `(x, y)` be an identity element.
`(a, b)*(x, y)=(a, b)`
`(ay+bx, by)=(a, b)`
`ay+bx=a=>x=0`
`by=b=>y=1`
Hence, `A` has no identity element.
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RD SHARMA-BINARY OPERATIONS-Solved Examples And Exercises
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  2. Let A=NxxN , and let * be a binary operation on A defined by (a ,\ ...

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  8. Let * be a binary operation on Q0 (set of non-zero rational numb...

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  9. Let * be a binary operation on Q-{-1} defined by a*b=a+b+a b for all...

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  10. Let * be a binary operation on Q-{-1} defined by a*b=a+b+a b for al...

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  11. Let R0 denote the set of all non-zero real numbers and let A=R0xxR0...

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  12. Let 'o' be a binary operation on the set Q0 of all non-zero rati...

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  13. On R-[1] , a binary operation * is defined by a*b=a+b-a b . Prove that...

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  14. Let R0 denote the set of all non-zero real numbers and let A=R0xxR0...

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  15. Let R0 denote the set of all non-zero real numbers and let A=R0xxR0...

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  16. Let * be the binary operation on N defined by a*b=H C F of a and b ...

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  17. Consider the set S={1,\ -1} of square roots of unity and multiplica...

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  18. Consider the set S={1,\ omega,\ omega^2} of all cube roots of unity...

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  19. Consider the set S={1,\ -1,\ i ,\ -i} of fourth roots of unity. Con...

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