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Let A be a non-empty set and S be the se...

Let `A` be a non-empty set and `S` be the set of all functions from `A` to itself. Prove that the composition of functions `'o'` is a non-commutative binary operation on `Sdot` Also, prove that `'o'` is an associative binary operation on `Sdot`

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`f,g in S`
`f:A->A`, `g:A->A`
Then , `fog:A->A` and `gof:A->A`
`(fog)(x)=f(g(x))` for all `x in A`
`implies fog in S`
Therefore o is binary operation.
To check commutative
Let `f,g in S`
...
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RD SHARMA-BINARY OPERATIONS-Solved Examples And Exercises
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  2. Consider the binary operations*: RxxR->R and o: RxxR->R defined as a...

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  3. Let A be a non-empty set and S be the set of all functions from A t...

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  5. Let A be a set having more than one element. Let * be a binary oper...

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  6. Let * be a binary operation on N defined by a*b=1. cdotmdot(a ,\ b)...

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  7. Let '*' be a binary operation on N given by a*b=LdotCdotMdot(a , b) fo...

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  8. Determine whether * on N defined by a*b=1 for all a ,\ b in N is asso...

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  9. Determine whether * on Q defined by a*b=(a+b)/2 for all a ,\ b in Q i...

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  10. Let A be any set containing more than one element. Let * be a binar...

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  11. Check the commutativity and associativity of * on Z defined by a*b=...

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  12. Check the commutativity and associativity of * on N defined by a*b=...

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  13. Check the commutativity and associativity of * on Q defined by a*b=...

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  14. Check the commutativity and associativity of O on Q defined by a\ \...

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  15. Check the commutativity and associativity of o on Q defined by a\ o...

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  16. Check the commutativity and associativity of * on Q defined by a*b=...

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  17. Check the commutativity and associativity of * on Q defined by a*b=...

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  18. Check the commutativity and associativity of * on R defined by a*b=...

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  19. Check the commutativity and associativity of * on Q defined by a*b=...

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  20. Check the commutativity and associativity of * on Q defined by a*b=...

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