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# Exam1Ans - MT004 MIDTERM 1 ANSWERS LKER S YCE QUESTION 1...

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MT004 MIDtERm 1 ANSWERS İLKER s. YÜCE May 11, 2010 QUEStION 1. seTs(15 PnTs) Suppose two dice are tossed and the numbers on the upper faces are observed. Let U denote the set of all possible pairs that can be observed. (These pairs can be listed, for example, by letting (2,3) denote a 2 was observed on the ﬁrst die and a 3 on the second.) Deﬁne the following subsets of U : A: The number on the second die is even. B: The sum of two numbers is even. C: At least one number in the pair is odd. List the points in A , C , A B , A B , A B and A C . ANSWER A = { (1 , 2) , (2 , 2) , (3 , 2) , (4 , 2) , (5 , 2) , (6 , 2) , (1 , 4) , (2 , 4) , (3 , 4) , (4 , 4) , (5 , 4) , (6 , 4) , (1 , 6) , (2 , 6) , (3 , 6) , (4 , 6) , (5 , 6) , (6 , 6) } , B = { (1 , 1) , (1 , 3) , (1 , 5) , (2 , 2) , (2 , 4) , (2 , 6) , (3 , 1) , (3 , 3) , (3 , 5) , (4 , 2) , (4 , 4) , (4 , 6) , (5 , 1) , (5 , 3) , (5 , 5) , (6 , 2) , (6 , 4) , (6 , 6) } C = { (1 , 1) , (1 , 2) , (1 , 3) , (1 , 4) , (1 , 5) , (1 , 6) , (2 , 1) , (2 , 3) , (2 , 5) , (3 , 1) , (3 , 2) , (3 , 3) , (3 , 4) , (3 , 5) , (3 , 6) (4 , 1) , (4 , 3) , (4 , 5) , (5 , 1) , (5 , 2) , (5 , 3) , (5 , 4) , (5 , 5) , (5 , 6) , (6 , 1) , (6 , 3) , (6 , 5) } C = { (2 , 2) , (2 , 4) , (2 , 6) , (4 , 2) , (4 , 4) , (4 , 6) , (6 , 2) , (6 , 4) , (6 , 6) } , A B = { (2 , 2) , (4 , 2) , (6 , 2) , (2 , 4) , (4 , 4) , (6 , 4) , (2 , 6) , (4 , 6) , (6 , 6) } , A B = { (1 , 2) , (3 , 2) , (5 , 2) , (1 , 4) , (3 , 4) , (5 , 4) , (1 , 6) , (3 , 6) , (5 , 6) } , A B = { (1 , 2) , (3 , 2) , (5 , 2) , (1 , 4) , (3 , 4) , (5 , 4) , (1 , 6) , (3 , 6) , (5 , 6) } , 1

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QUESTION 2. A FunDAMentAL PRinCiPLe oF CountinG(15 Pnts) A. Let U denote a universal set with 54 elements and S and T be two subsets of U so that n ( S T ) = 2 × n ( S T ) and n ( S T ) = 2 × n (( S T ) ( S T )) and n (( S T ) ) = 3 × n ( S T ) . Draw a two circle Venn diagram and ﬁnd the number of elements in each basic region.
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## This note was uploaded on 02/23/2011 for the course MATH 444 taught by Professor Any during the Fall '10 term at Roosevelt.

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Exam1Ans - MT004 MIDTERM 1 ANSWERS LKER S YCE QUESTION 1...

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