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Let R0 denote the set of all non-zero...

Let `R_0` denote the set of all non-zero real numbers and let `A=R_0xxR_0` . If * is a binary operation on `A` defined by `(a ,\ b)*(c ,\ d)=(a c ,\ b d)` for all `(a ,\ b),\ (c ,\ d) in Adot` Show that * is both commutative and associative on `A` (ii) Find the identity element in `A`

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Let `(a, b)` and `(c, d) in` A for all `a, b, c, d in R_{0}`. Then,
`(a, b) *(c, d)=(a c, b d) `
`=(c a, b d) `
`=(c, d) *(a, b) `
`therefore(a, b) *(c, d)=(c, d) *(a, b)`
Thus, `*` is commutative on `A`
Associativity:
...
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RD SHARMA-BINARY OPERATIONS-Solved Examples And Exercises
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  2. Let * be a binary operation on Q-{-1} defined by a*b=a+b+a b for al...

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

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

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

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

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

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  12. Consider the set S={1,\ 2,\ 3,\ 4} . Define a binary operation * on...

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  13. \begin{tabular}{|l|l|l|l|l|l|} \hline 1 & 1 & 2 & 3 & 4 & 5 \\ \hline ...

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  14. Consider a binary operation * on the set {1, 2, 3, 4, 5} given by t...

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  15. Define a binary operation * on the set A={0,\ 1,\ 2,\ 3,\ 4,\ 5} as...

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  16. Define a binary operation * on the set A={0,1,2,3,4,5} as a*b=a+b (mod...

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  17. Define a binary operation * on the set A={1,\ 2,\ 3,4} as a*b=a b\ ...

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  19. Construct the composition table for xx4 on set S={0,\ 1,\ 2,\ 3} .

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