lec20-review-exam2 - EECS 203 Winter 2007 Discrete...

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EECS 203, Winter 2007 Discrete Mathematics Lecture 20 Review for the 2nd Exam March 20 Reading: Covered sections in Rosen March 20 Review for the 2nd Exam, Page 1
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20.1 Matrices and graphs Key concepts: matrix, symmetric, transpose, matrix multiplication, graphs (undirected and directed), path, cycle, walks, connected, connected components, isomorphism, subgraphs, adjacency matrix, incidence matrix. Key results: If A is the adjacency matrix of G , then [ A k ] u,v is the number of different walks from u to v using exactly k steps. Ramsey’s Theorem: for any integers s and t , there exists R ( s, t ) so that any graph with at least R ( s, t ) vertices would have a size s clique, or its complement has a size t clique. March 20 Review for the 2nd Exam, Page 2
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20.2 Sum and sequence i a i , i ia i - 1 i i , i i 2 , i i ( i - 1). Approximation by integration: ln( n + 1) = n +1 x =1 1 /xdx n i =1 1 i 1 + n x =1 1 /xdx = 1 + ln n. An approximation of n !: n x =1 ln xdx ln n ! = i ln i n +1 x =2 ln xdx. Since ln xdx = x ln x - x + C , n ln n e ln n ! ( n + 1) ln n + 1 e . March 20 Review for the 2nd Exam, Page 3
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20.2 Sum and sequence A better bound 2 πn ( n e ) n e 1 12 n +1 < n ! < 2 πn ( n e ) n e 1 12 n . Stirling’s formula:
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