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Exam3-Solutions

# Exam3-Solutions - 1(15 points 3 points each Let A = cfw_4 5...

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1. (15 points, 3 points each) Let A = { 4 , 5 , 6 } , B = { 3 , 4 } , C = { 7 , 8 , 9 } . Find (a) A C = { 4 , 5 , 6 , 7 , 8 , 9 } (b) A - B = { 5 , 6 } (c) A × B = { (4 , 3) , (4 , 4) , (5 , 3) , (5 , 4) , (6 , 3) , (6 , 4) } (d) A B = { 4 } (e) A C = 2. (15 points, 5 points each part) (a) Find the Power Set of X = { a,b } {∅ , { a } , { b } , { a,b }} (b) Transform the following by making a change of variable i = k + 1: n X k =0 k 2 k + n n +1 X i =1 ( i - 1) 2 i - 1 + n (c) Use summation notation to rewrite the following: 1 - 1 2 + 1 3 - 1 4 + 1 5 - 1 6 6 X k =1 ( - 1) k +1 k 3. (15 points, 3 points each part) Give concise deﬁnitions of the following terms (a) Well ordering principle: Let S be a set containing one or more integers all of which are greater than some ﬁxed integer. Then S has a least element. (b) Cartesian Product: The Cartesian Product A × B of two sets is the set of all ordered pairs ( a,b ) such that a A and b B , A × B = { ( a,b ) | a A,b B } (c) Ordered n-tuple: An ordered n - tuple ( x 1 ,x 2 ,...,x n ) is a collection of not- necessarily distinct elements of some set X together with an ordering such that x 1 comes before x 2 , etc., in the tuple. (d) Disjoint sets:

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Exam3-Solutions - 1(15 points 3 points each Let A = cfw_4 5...

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