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Exercises

# Exercises - Exercise Sheet 1 1 Let A B be sets What does...

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Exercise Sheet 1 1) Let A, B be sets. What does the statement "A is not a subset of B " mean? 2) Let A, B, C, X be sets with A, B, and C are subsets of X. Prove the following set equalities a) (A B)' = A' B' b) (A B)' = A' B' where the complements are taken in X. c) A \ B = A B' d) A (B \ C) = (A B)\(A C) 3) Let A = {x R | x 2 -5x+4 0}; B = {x R | |x 2 - 1/2| 1} and C = {x R | x 2 -7x+12 < 0}. Determine (A B) C. 4) Let f: X Y be a mapping; Let A, B X; C, D Y. Prove that : a) f(A B) f(A) f(B); Find examples of A, B, and f such that f(A B) f(A) f(B) b) f -1 (C D) = f -1 (C) f -1 (D) 5) Let f : R \ {0} R and g : R R x | 1/x x | 2x/(1+x 2 ) be mappings. a) Determine f g and g f. b) Find the image g( R). Is g injective? surjective? (Answer the same question for f. ) 6) Let f : A C and g : B D be two mappings. Consider the mapping h : A x B C x D defined by h(a, b) = (f(a), g(b)) for all (a, b) A x B. a) Prove that, f and g are both injective if and only if h is injective. b) Prove that, f and g are both surjective if and only if h is surjective.

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Exercise Sheet 2
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Exercises - Exercise Sheet 1 1 Let A B be sets What does...

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