# hmk2_key - COT 3100 Spring2001 Homework #2 Assigned: Jan....

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COT 3100 Spring’2001 Homework #2 Assigned: Jan. 24/25 Due: Feb. 07/08 in recitation 1. Let A be the set of prime numbers, let B be the set of positive even integers, and let C be the set of positive multiples of 3. The universal set is the set of positive integers. Determine the following sets: list all of their elements if they have 10 or fewer elements; list their 10 smallest elements if they have more than 10. a) A C = {3} b) ( C B ) – A = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …} c) ¬ ( B C ) = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, …} d) ( B C ) ( C B ) = {2, 3, 4, 8, 9, 10, 14, 15, 16, 20, …} e) A ( B C ) = {2, 3, 4, 5, 7, 8, 10, 11, 13, 14, …} 2. (p. 138, # 12). For U = Z + let A U where A = {1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 17, 18}. a) How many subsets of A contain six elements? This is the number of combinations of size 6 from 12, C (12, 6)=12!/(6! 6!)=924. b) How many six-element subsets of A contain four even integers and two odd integers? We can choose four even integers from 6 available {2, 4, 8, 10, 14, 18} in C (6, 4)= 6!/(4! 2!)=15 different ways. Two odd integers can be picked from six available in A in C(6, 2)=15 different ways. By the product rule the number of six element subsets with four even and two odd is 15 15=225. c) How many subsets of A contain only odd integers? That is the number of subsets of a six-element set 2 6 =64. 3. Let A and B be arbitrary sets. Prove that a) ( A B ) = ( A B ) if and only if A = B Proof . We need to prove that two propositions are identical, i.e. ( A B ) = ( A B ) A = B . That is equivalent to prove two implications ( A B ) = ( A B ) A = B and A = B ( A B ) = ( A B ). Part 1.

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hmk2_key - COT 3100 Spring2001 Homework #2 Assigned: Jan....

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