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# exercises_4 - Y and A ⊂ Y B ⊂ Y Prove that(a f-1 A ∪...

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Analysis 1: Exercises 4 1. Let f : X Y and g : Y Z be functions. Prove that (a) if f and g are both injective then g f is injective; (b) if f and g are both surjective then g f is surjective. 2. Let f : X Y and g : Y Z be functions. (a) If g f is surjective, show that g is surjective. (b) If g f is surjective and g is injective, show that f is surjective. 3. Let f : R - {- 1 , 1 } → R - { 0 } defined by f ( x ) = x +1 x - 1 , g : R - { 0 } → R defined by g ( x ) = 1 x . Determine g f . 4. Let X = R - { 1 } . Let f : X X be defined by f ( x ) = x + 1 x - 1 . Prove that (a) f is a bijection. (b) f f = i X . ( i X is the identity mapping in X , i.e. ( x X )[ i X ( x ) = x ]). (c) What is the inverse function of f ? 5. Let f : R - { 1 } → R defined by f ( x ) = 3 x x - 1 . (a) Prove that f is an injection. (b) Find the range Ran ( f ). (c) Define the function ˜ f : R - { 1 } → Y defined by ˜ f ( x ) = 3 x x - 1 with Y = Ran ( f ). Show that ˜ f is a bijection and find the inverse finction ˜ f - 1 . 6. Let
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Unformatted text preview: Y and A ⊂ Y , B ⊂ Y . Prove that (a) f-1 ( A ∪ B ) = f-1 ( A ) ∪ f-1 ( B ), (b) f-1 ( A ∩ B ) = f-1 ( A ) ∩ f-1 ( B ). 7. Let X and Y be ﬁnite sets with m elements in X and n elements in Y . (a) How many functions are there from X to Y ? (b) How many injections are there from X to Y ? (c) How many bijections are there from X to Y ? 8. Prove that (a) ( ∀ n ∈ N + ) 1 3 + 2 3 + 3 3 + ··· + n 3 = ± n ( n + 1) 2 ² 2 . (b) ( ∀ n ∈ N + ) 1 · 1! + 2 · 2! + 3 · 3! + ··· + n · n ! = ( n + 1)!-1. (c) ( ∀ n ∈ N + ) (64 | (9 n-8 n-1)). (d) ( ∀ n ∈ N + ) (19 | (5 · 2 3 n-2 + 3 3 n-1 ))....
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