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Unformatted text preview: 550.171 Discrete Mathematics  Fall 2003 Midterm #1
Print your name on. each. sheet; No calculators; No notes; Have your ID ready for viewing. Name ___________________________________________________ ID# ______________________________ 1. Let AB and C he sets. Label each of the following statements as either “TRUE” or “FALSE”
(not both], and support your conclusion with a Venn diagram of the situation. Then disprove the false 0116(5). You do not need to prove the true ones. (a) [10 pts] (A as) — (A — B): B (b) [lUpts]AU(B~—C)=(AUB)—(BHC) Name ______________________________________________________ 2. {15 pts] Use truth tables to prove that a: <—> y is logically equivalent to (a: —> y) A ((ﬁz) —+ (—11.0). 3. Parts (8.) and (b) of this problem are entirely separate questiOns.
(a) [5 pt} Let x and y be integers. We say a: is of the same parity as y provided as — y is an even
integer. Let R be the isof—thesameparityas relation on the set {1,2,314}. List all the elements of R. (b) [5 pt] How many possible relations on a set with n elements are there? Show your work and
explain what you did. Name ______________________________________________________ 4. (a) £5 pt] Write a careful deﬁnition of what it means for a positive integer to be triangular. For
example, 1, 3=1+2, 6=1+2+3 and 10=1+2+3+4 are the ﬁrst four triangular numbers. {13) [20 pt] For integer n 2 1, let Tfl denote the nth triangular number (so, T1 : 1,T2 : 3,33 = 6,
etc.). Prove the following statement: ' Ifn 2 3 is an integer, then T11 — Tﬂ_2 is an odd integer. Name ____________________________________________________ 5. A classroom has 20 children comprised of 10 boys and 10 girls. Leave all your answers in product
form ~ do not simplify. (a) [5 pt] If the children are to walk out of the room all in a row, how many different arrangements
can be observed '? (b) [10 pt] Same question as in part (a) except that all children of the same gender must walk out
together; i.e., either all 10 girls follow the 10 boys, or all 10 boys follow the 10 girls. 13! 6. (a) [6 pt] Compute (12b 10
H %. Show your work, and simplify your answer fully for full credit.
k=1 (b) [3 pt each} Label as TRUE or FALSE not both. Please spell the word out completely.
(1) _________i 0! = 0
(2) ____s______ If A, B and C are pairwisedisjoint sets, then A H B O C = G (3) ____________ It is possible for the cardinality of the power set of some set to be one. ...
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 Spring '10
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