171fa04_midterm1

# 171fa04_midterm1 - for content and correctness. Name 4. [4...

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550.171 Discrete Mathematics - Fall 2004 Midterm #1 Name Print your name on each sheet; No Calculators; No notes; Leave your ID on your desk. 1. Short Answer. (a) [6 pt] State the deﬁnition of a relation on a set A . (b) [7 pt] Let A ∈ { 1 , 2 , 3 } . How many relations on the set A are reﬂexive? 2. A baseball team has 9 players, one for each of the following positions: pitcher, catcher, ﬁrst base, second base, third base, shortstop, rightﬁeld, centerﬁeld, and leftﬁeld. I have twenty players on the bench. (Leave your answer unsimpliﬁed, e.g. do not write 120 even though your answer may be 5!) (a) [8 pt] How many diﬀerent baseball teams can be formed from these 20 players. Consider it the same team if the same 9 people are playing in diﬀerent positions. (b) [7 pt] How many ways can I assign positions to each of the 20 players on the bench. For example, I could make all 20 players shortstops.

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Name 3. [25 pt] Provide a direct proof of the following statement: If x 1 is an integer, then x 2 + 2 x + 1 is composite. Your proof will be graded as much on style, grammar and use of complete sentences, as it will

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Unformatted text preview: for content and correctness. Name 4. [4 pt each] TRUE or FALSE (spell out the word) (a) A for any set A . (b) If A and B are nite sets, then | A B | = | A | | B | (c) If R is an equivalence relation on a set A , then R cannot be antisymmetric. (d) x Z , y Z , ( x + y ) | (e) For every x N , n Y k =1 (2 k ) = 2 n n ! 5. [15 pt] Prove or disprove: If A,B and C are sets such that A B C , then A B C . Name 6. [15 pt] Use a truth table proof to prove that x ( x y ) is logically equivalent to x y . 7. [7 pt] Let R be a symmetric relation on on a nite set A that is irreexive. Briey explain why R must have an even number of elements. Hint: if A is not empty, then R has no ordered pairs of the form ( x,x ). The only other kind of ordered pair allowed in R will then be of the form ( x,y ). But, R is symmetric!...
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## This note was uploaded on 03/19/2010 for the course EOE 342 taught by Professor Jooe during the Spring '10 term at Albany College of Pharmacy and Health Sciences.

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171fa04_midterm1 - for content and correctness. Name 4. [4...

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