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Let f: X rarr Y If there exists a map ...

Let f: `X rarr Y`
If there exists a map g:Y `rarr` X such that gof = `id_(X)` and fog = `id_(y)`, then show that
f is bijective and (ii) `g =f^(-1)`

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SHARAM PUBLICATION-RELATIONS AND FUNCTIONS-EXAMPLE
  1. Prove that f:X rarr Y is surjective iff for all B sube Y, f(f^(-1)(B))...

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  2. Prove that for any f:X rarr Y , f o idx = f =idY of.

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  3. Let f: X rarr Y If there exists a map g:Y rarr X such that gof = id...

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  4. Let f:XrarrY. If there exists a map g:YrarrX such that g of=idxand fo ...

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  5. If ff(x)=cos[pi^2]x+cos[-pi^2]x where [x] stands for the greatest inte...

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  6. If f:RrarrR,g :RrarrR and h : RrarrR such that f(x)=x^2, g(x)= tan x a...

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  7. If p is a prime and ab-=0 (mod p) then show that either a=0 (mod p) or...

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  8. Prove that the relation R on the set Z of all integers defined by R={(...

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  9. Let n be positive integer and a function f be defined as f(n)={(0 , wh...

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  10. If f:RrarrR defined by f(x)=5x-8 for all x inR, then show that f is in...

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  11. Show that the inverse of a bijective function is unique.

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  12. Show that the inverse of a bijective is also a bijection.

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  13. Let f={(1,a),(2,b),(3,c),(4,d)} and g={(a,x),(b,x),(c,y),(d,x)} Determ...

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  14. Prove that the greatest integer function f:R rarr R, given by f(x) = [...

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  15. Let A and B be sets. Show that f : A xx B rarr B xx A such that f (...

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  16. Show that the fuction f:RrarrR defined by f (x)=sin x is neither one-...

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  17. If f:N to N is defined by. f(n)={((n+1)/(2)", if n is odd"),((n)/(2)...

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  18. Show that a fuction f:RrarrR given by f(x)=ax+b, a, bin R and a!=0 is ...

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  19. Let* be a binary operation on Q, defined by a*b=(3ab)/(5) .Show that i...

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  20. If A=N xx N and * is a binary operation on A defined by (a, b) ** (c,...

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