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Prove that f:X rarr Y is injective iff f...

Prove that `f:X rarr Y` is injective iff `f^(-1) (f(A)) = "A for all" A sube X`.

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Prove that f: X rarr Y is surjective iff for all A sube X,(f(A))' sube f(A') , where A' denotes the complement of A in X.

Prove that f:X rarr Y is injective iff for all subsets A,B of X,f(A nn B ) = f(A) nn f(B) .

Prove that f:X rarr Y is surjective iff for all B sube Y, f(f^(-1)(B)) =B.

Prove that f:X to Y is injective iff for all subsets A, B of X, f (A cap B) =f(A) cap f(B) .

If f:XrightarrowY is onto, then prove that f(f^(-1)(B))=B for all BsubseteqY .

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Show that the function f: N rarr N, given by f(1) f(2)= 1 and f(x) = x - 1 for every x gt 2, is onto but not one-one.

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SHARAM PUBLICATION-RELATIONS AND FUNCTIONS-EXAMPLE
  1. if ** is the binary operation on N given by a**b= L. C. M of a and b. ...

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  2. Prove that f:X to Y is injective iff for all subsets A, B of X, f (A c...

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  3. Prove that f:X rarr Y is injective iff f^(-1) (f(A)) = "A for all" A s...

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  4. Prove that f:X rarr Y is surjective iff for all B sube Y, f(f^(-1)(B))...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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