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Let `X` be a non-empty set and let * be a binary operation on `P(X)` (the power set of set `X)` defined by `A*B=AuuB` for all `A ,\ B in P(X)` . Prove that * is both commutative and associative on `P(X)` . Find the identity element with respect to * on `P(X)` . Also, show that `varphi in P(X)` is the only invertible element of `P(X)dot`

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`A,B in P(X)`
Commutative property
`A*B=A cup B` and `B*A=B cup A`
`A*B=B*A` for all `B, A in P(X)`
Associative property
`(A*B)*C=(A cup B)*C=>A cup B cup C`
`A*(B*C)=A*(B cup C)=> A cup B cup C`
`(A*B)*C=A*(B*C)` ...
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RD SHARMA-BINARY OPERATIONS-Solved Examples And Exercises
  1. Define a binary operation * on the set A={0,1,2,3,4,5} given by a*b=a ...

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  2. On the set M=A(x)={[xxxx]: x in R}of2x2 matrices, find the identity ...

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  3. Let X be a non-empty set and let * be a binary operation on P(X) (t...

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  4. Let X be a nonempty set and *be a binary operation on P(X), the power ...

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  5. Let X be a non-empty set and let * be a binary operation on P\ (X) ...

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  6. Let X be a non-empty set and let * be a binary operation on P\ (X) ...

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  7. Let A=QxxQ and let ** be a binary operation on A defined by (a ,\ b)...

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  8. Let A=QxxQ and let * be a binary operation on A defined by (a ,\ b)...

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  9. Let A=Nuu{0}xxNuu{0} and let * be a binary operation on A defined b...

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  10. Let A=Nuu{0}xxNuu{0} and let * be a binary operation on A defined b...

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  11. Let A=NxxN , and let * be a binary operation on A defined by (a ,\ ...

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  12. Let A=NxxN , and let * be a binary operation on A defined by (a ,\ ...

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  13. Show that the number of binary operations on {1," "2} having 1 as iden...

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  14. Determine the total number of binary operations on the set S={1,\ 2...

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  15. Let * be a binary operation on Z defined by a*b=a+b-4 for all a ,\ ...

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  16. Let * be a binary operation on Z defined by a*b=a+b-4 for all a ,\ ...

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

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

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

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

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