4023f10ex1a

# 4023f10ex1a - Name: Exam 1 Instructions. Answer each of the...

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Name: Exam 1 Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [ 15 Points] Let T = f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g . Give the number of elements in each of the following sets. (a) The cartesian product T £ T . I Solution. j T £ T j = j T j 2 = 6 2 = 36. J (b) The power set P ( T ). Recall that P ( X ) is the set of all subsets of X . I Solution. jP ( T ) j = 2 j T j = 2 6 = 64 : J (c) The set P 3 ( T ) consisting of all 3-element subsets of T . I Solution. jP 3 ( T ) j = ¡ 6 3 ¢ = 6! 3!(6 ¡ 3)! = 6 ¢ 5 ¢ 4 3 ¢ 2 ¢ 1 = 20 : J (d) P ( T ) [ T . I Solution. jP ( T ) [ T j = jP ( T ) j + j T j ¡ jP ( T ) \ T j = 64 + 6 ¡ 0 = 70 : Note that jP ( T ) \ T j = 0 since jP ( T ) \ T j = ; . A possible confusion that led some people to a wrong conclusion is the assumption that the element a 2 T is also an element of P ( T ) in the form of the singleton subset f a g of T . However a 2 T and f a g ‰ T are diﬁerent objects; one is an element of T and one is a subset of T containing a single element. Namely,

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4023f10ex1a - Name: Exam 1 Instructions. Answer each of the...

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