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On R-[1] , a binary operation * is defin...

On `R-[1]` , a binary operation * is defined by `a*b=a+b-a b` . Prove that * is commutative and associative. Find the identity element for * on `R-[1]dot` Also, prove that every element of `r-[1]` is invertible.

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Commutativity:
Let `a, b in R-{1}`. Then,
`a * b=a+b-a b `
`quad=b+a-b a `
`quad=b * a`

Therefore, ...
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RD SHARMA-BINARY OPERATIONS-Solved Examples And Exercises
  1. Let R0 denote the set of all non-zero real numbers and let A=R0xxR0...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  16. Construct the composition table for the composition of functions (o...

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

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  18. Construct the composition table for +5 on set S={0,\ 1,\ 2,\ 3,\ 4}...

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  19. Construct the composition table for xx6 on set S={0,\ 1,\ 2,\ 3,\ 4...

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  20. Construct the composition table for xx5 on Z5={0,\ 1,\ 2,\ 3,\ 4} .

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