171fa04_midterm2 - 550.171 Discrete Mathematics Fall 2004...

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550.171 Discrete Mathematics - Fall 2004 Midterm #2 Name Print your name on each sheet; No Calculators; No notes; Leave your ID on your desk. 1. Let A = { a,b,c,d } and R = n ( a, { b,c,d } ) , ( b, { a,c,d } ) , ( c, { a,b,d } ) , ( d, { a,b,c } ) o . (a) [2 pt] Is R a function? Yes or No (circle one) (b) [3 pt] If I = n B A : | B | = 3 o , which if the following are true: (circle all that apply) (I) R : A → I (II) R : A → I is one-to-one (III) R : A → I is onto (c) [3 pt] How many functions f : A 2 A are there? Recall: 2 A represents the power set of A . (d) [2 pt] How many functions f : A → I are one-to-one? Recall I is defined in part (b) of this problem. 2. [15 pt] In the very first day of class we learned the following definition: Let a and b be integers. We say a divides b (written a | b ) provided there exists an integer x such that b = ax . Prove that if a 6 = 0, then this integer x is unique.
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Name 3. [25 pt] Prove: For every integer n 4, n 2 2 n .
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Name 4. Short answer: (a) [4 pt] State the contrapositive of the statement: If A and not B , then C or D .
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171fa04_midterm2 - 550.171 Discrete Mathematics Fall 2004...

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