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Practice Quiz 3 Recursive Definitions Relations Basic Counting Pigeonhole Principle Permutations & Combinations Discrete Probability

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Problems The Fibonacci numbers f 0, f 1, f 2, … are defined as f 0 = 0, f 1 = 1, and f n = f n-1 + f n-2, for n > 1. Prove that the function f(n) = f 1 + f 3 + … + f 2n-1 is equal to f 2n whenever n is a positive integer.

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Consider the relation R = {x, y | x + y > 10} on the set of positive integers. –Is R reflexive? Justify your answer. –Is R symmetric? Justify your answer. –Is R antisymmetric? Justify your answer. –Is R transitive? Justify your answer. Suppose that R1 and R2 are symmetric relations on a set A. Prove or disprove that R1 – R2 (set difference) is also symmetric. Problems

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Given the relation on the set {0, 1, 2, 3} defined by the ordered pairs (0,0), (1,1), (1,3), (2,2), (2,3), (3,1), (3,3) –What is the 0-1 matrix representation of this relation? –Is the relation reflexive, symmetric and/or transitive? –Is the relation an equivalence relation?

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How many students must be in a class to guarantee that at least 5 were born on the same day of the week? What is the coefficient of x 7 y 12 in the expansion of (x+y) 19 ? What is the probability that a randomly selected day of the year (365 days) is in May? Suppose that you pick two cards one at a time at random from an ordianary deck of 52 cards. Find p(both cards are diamonds). Problems

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The Fibonacci numbers, f 0, f 1, f 2, … are defined as f 0 = 0, f 1 = 1, and f n = f n-1 + f n-2, for n > 1. Prove that the function (n) = f 1 + f 3 + … + f 2n-1 is equal to f 2n whenever n is a positive integer. Basis Step If n = 1, then (1) = f 2*1 - 1 = f 1 = 1 = f 2.

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Inductive Step Assume (k) = f 1 + f 3 + … + f 2k-1 = f 2k for k n. We must show that this implies that (n+1) = f 2(n+1) (n+1) = f 1 + f 3 + … + f 2n-1 + f 2n+1 (n+1) = (n) + f 2n+1 (n+1) = f 2n + f 2n+1 = f 2n+2 = f 2(n+1) The Fibonacci numbers, f 0, f 1, f 2, … are defined as f 0 = 0, f 1 = 1, and f n = f n-1 + f n-2, for n > 1. Prove that the function (n) = f 1 + f 3 + … + f 2n-1 is equal to f 2n whenever n is a positive integer.

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Given the relation on the set {0,1,2,3} defined by the ordered pairs: (0,0), (0,2), (1,1), (1,2), (2,0), (2,2), (3,3). Is the relation: – Reflexive? Yes –Symmetric? No –Transitive? No –An equivalence relation? No What is the 0-1 matrix?

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Suppose that R1 and R2 are symmetric relations on a set A. Prove or disprove that R1 – R2 (set difference) is also symmetric. Proof: Assume that R1 and R2 are symmetric relations on a set A. We must show that if (a,b) R1 - R2, then (b,a) R1 – R2. If (a,b) R1 – R2, then (a,b) R1 and (a,b) R2. Since R1 is symmetric, (b,a) R1. Since R2 is symmetric, it follows that (b,a) R2, for if (b,a) was in R2, then (a,b) would have been in R2. Hence, since (b,a) is in R1 and not in R2, (b,a) R1 – R2. Therefore R1 – R2 is symmetric.

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Consider the relation R = {x, y | x + y > 10} on the set of positive integers. Is R reflexive? Justify your answer. Is R symmetric? Justify your answer. Is R antisymmetric? Justify your answer. Is R transitive? Justify your answer. R is not reflexive, since 1 + 1 < 10, so (1,1) is not in R. R is symmetric, since x + y > 10 means that y + x > 10, so (x, y) R implies (y, x) R. R is not antisymmetric, since (2, 10) and (10, 2) R, but 2 10. R is not transitive since (2, 9) R and (9, 3) R, but (2, 3) R because 2 + 3 < 10.

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How many students must be in a class to guarantee that at least 5 were born on the same day of the week? By the pigeonhole principle: 7*4+1 = 29

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What is the coefficient of x 12 y 7 in the expansion of (x+y) 19 ? Solution: Use the binomial expansion formula So the 8th coefficient is C(19,7) = 19!/7!12! = 50,388.

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