HW3_199

# HW3_199 - B(a(1 point Show that f ∪ i ∈ I A i = ∪ i...

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HOMEWORK 3 DUE: Fri., Oct. 14 NAME: DIRECTIONS: Turn in your homework as SINGLE-SIDED typed or handwritten pages. STAPLE your homework together. Do not use paper clips, folds, etc. STAPLE this page to the front of your homework. Be sure to write your name on your homework. Show all work, clearly and in order . You will lose point 0.5 points for each instruction not followed. Questions Points Score 1 1 2 1 3 1 4 1 5 2 6 1 7 2 8 1 Total 10 1

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HOMEWORK 3 DUE: Fri., Oct. 14 NAME: Problem 1: (1 point) (a) (0.5 point) Show that { ( a, b ) } is an additive inverse for { ( a, b ) } . (b) (0.5 point) Prove the distributive law for Q . Problem 2: (1 points) Let R be a ring and R 0 a nonempty subset of R . Show that R 0 is a subring iF, for any a , b R 0 , we have a b , ab R 0 . Problem 3: (1 point) Let A = { p, q, r } and B = { π, e } . Determine all possible functions from A to B . Problem 4: (1 points) Given f : A B , suppose there exist g , h : B A so that f g = I B and h f = I A . Show that f is a bijection and that g = h = f 1 . Problem 5: (2 points) Let A and B be sets and let f : A B be a function. Suppose that { A i } i I is a collection of subsets of A and { B j } j J is a collection of subsets of
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Unformatted text preview: B . (a) (1 point) Show that f ( ∪ i ∈ I A i ) = ∪ i ∈ I f ( A i ). (b) (1 point) Suppose f is a bijection. Show that f − 1 ( ∩ i ∈ J B j ) = ∩ j ∈ J f − 1 ( B j ). Problem 6: (1 points) ±or following functions, ²nd f ( A ) and f − 1 ( B ). (a) (0.5 point) f : R → R is de²ned by f ( x ) = sin x , A = {− 2 , − 1 , , 1 , 2 } , B = { , 1 , 2 } . (b) (0.5 point) f : R → Z is is the ³oor function de²ned by f ( x ) = ⌊ x ⌋ = n where n ≤ x < n + 1 for n ∈ N and A = (0 , 5), B = { , 1 , 2 } . Problem 7: (2 points) Let f : A → B and B ′ ⊆ B . (a) (1 point) Prove that f ( f − 1 ( B ′ ) ) ⊆ B ′ . (b) (1 point) Prove that if f is onto, then f ( f − 1 ( B ′ ) ) = B ′ . Problem 8: (1 point) Prove or ²nd a counterexample to the following conjecture. Assume f : X → Y and A, B ⊂ X If f ( A ) \ f ( B ) = ∅ , then f ( A \ B ) = ∅ . 2...
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HW3_199 - B(a(1 point Show that f ∪ i ∈ I A i = ∪ i...

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