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Prove that **: R xx R to R defined as a...

Prove that `**: R xx R to R ` defined as `a ** b =a+2ab` is not commutative .

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To prove that the operation defined as \( a \ast b = a + 2ab \) is not commutative, we need to show that \( a \ast b \neq b \ast a \) for at least one pair of real numbers \( a \) and \( b \). ### Step-by-step Solution: 1. **Define the operation**: We have the operation defined as: \[ a \ast b = a + 2ab ...
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NAGEEN PRAKASHAN ENGLISH-RELATIONS AND FUNCTIONS -Miscellaneous Exercise
  1. Prove that **: R xx R to R defined as a ** b =a+2ab is not commutativ...

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  2. Let f: R ->Rbe defined as f(x) = 10 x + 7. Find the function g: R ->R...

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  3. Let f: W ->Wbe defined as f(n) = n - 1, if is odd and f(n) = n + 1, i...

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  4. If f: R ->Ris defined by f(x) = x^2- 3x + 2, find f(f(x)).

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  5. Show that the function f: R->{x in R :-1ltxlt1} defined by f(x)=x/(1+...

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  6. Show that the function f: R->Rgiven by f(x)=x^3is injective.

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  7. Give examples of two functions f: N->Z and g: Z->Z such that gof is...

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  8. Given examples of two functions f:" "N ->N" "a n d""""""g:" "N->N such...

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  9. Given a non-empty set X, consider P(X) which is the set of all subsets...

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  10. Given a non -empty set X, let *:" "P(X)" "xx" "P(X) ->P(X) be defined ...

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  11. Find the number of all onto functions from the set A={1,\ 2,\ 3,\ ...

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  12. Let S = {a , b , c} a n d T = {1, 2, 3}. Find F^(-1)of the following ...

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  13. Consider the binary operations*: RxxR->R and o: RxxR->R defined as a...

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  14. Given a non -empty set X, let *:" "P(X)" "xx" "P(X) ->P(X) be defined ...

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

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  16. Let A" "=" "{-1," "0," "1," "2} , B" "=" "{-4," "-2," "0," "2} and f,g...

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  17. Let A={1,\ 2,\ 3} . Then, the number of relations containing (1, 2) ...

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  18. Let A = {1, 2, 3}. Then number of equivalence relations containing (1...

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  19. Let f: R->Rbe the Signum Function defined as f(x)={1,x >0 0,x=0-1,x<1 ...

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  20. Number of binary operations on the set {a, b} are (A) 10 (B) 16 (C)...

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