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Assignment5

# Assignment5 - ± on S by a ± b if a | b Draw the Hasse...

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Math 240, Fall 2009 — Assignment 5 — Due: Monday, November 30 1. (Peer reviewed question) Let f n denote the n th Fibonacci number (deﬁned recursively by f 0 = 0, f 1 = 1, and f n +2 = f n +1 + f n whenever n 0). Prove that f n +1 f n - 1 - ( f n ) 2 = ( - 1) n whenever n is a positive integer. 2. Solve the recurrence a n = a n - 1 + 6 a n - 2 , with initial conditions a 0 = 7, a 1 = 1. 3. Use a generating function to solve the recurrence relation a k = 3 a k - 1 + 4 k - 1 with the initial condition a 0 = 1. 4. Here are some relations R which are deﬁned on a set A . Are they equivalence relations? Are they partial orders? Explain. (a) A is the set of all rooted trees whose nodes are distinct elements of the set { a,b,c,d,e,f } , and aRb if a appears as a subtree of b . (b) A is the set of ordered pairs of positive integers, and ( a,b ) R ( c,d ) if ad = bc. 5. (a) Let S = { 1 , 2 , 3 ,..., 16 } . Deﬁne a partial order

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Unformatted text preview: ± on S by a ± b if a | b . Draw the Hasse diagram for the partially ordered set ( S, ± ). Is ( S, ± ) a lattice? (b) Now replace S with the set { 3 a 7 b | ≤ a,b ≤ 3 } , and use the same ordering ± as in part (a). Draw the Hasse diagram for the partially ordered set ( S, ± ). Is ( S, ± ) a lattice? 6. Let G be a graph with v vertices and e edges (and suppose that v > 0). Let M be the maximum degree of the vertices of G , and let m be the minimum degree of the vertices of G . Show that (a) 2 e/v ≥ m . (b) 2 e/v ≤ M . Math 240 — Assignment 5 — Fall 2009 Marker: Benjamin Young Instructor: Benjamin Young Date: Name: Student Number:...
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Assignment5 - ± on S by a ± b if a | b Draw the Hasse...

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