final-sol - Math 55-2 Final Exam SOLUTIONS Rob Bayer August...

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Math 55-2 Final Exam SOLUTIONS Rob Bayer August 14, 2009 You have until 4:00pm to complete this test. No calculators, books, notes, or consultation with other members of the class are permitted. Your exam should have 4 pages. Unsupported or improperly supported answers will receive no credit. 1. (10 pts) Short Answer. You need not show any work for this problem (a) A function f is injective iff f ( x ) = f ( y ) x = y (b) A relation R is an equivalence relation iff it is symmetric, transitive, reflexive (c) Two events are called independent iff P(E —F) = P(E) (d) Circle the countable sets: R Q Z × N P ( N ) (e) The linear congruence ax b (mod n ) has a solution iff gcd( a,n ) | b (f) A function f : G 1 G 2 is an isomorphism of graphs iff f is bijective and there is an edge b/w u and v in G 1 iff there is an edge b/w f ( u ) and f ( v ) in G 2 (g) State Chebyshev’s Inequality P ( | X - E [ X ] | ≥ r ) V ( X ) /r 2 (h) Find a Minimal Spanning Tree in the following graph. You may simply fill in the necessary edges graph-sol.pdf (i) Find a generating function for the number of solutions in non-negative integers to 2 x 1 + 3 x 2 + 4 x 3 = n 1 1 - x 2 1 1 - x 3 1 1 - x 4 2. (15 pts) (a) Find a truth table for the compound proposition ( p q ) ( q ∧ ¬ r ) p q r p q q ∧ ¬ r ( p q ) ( q ∧ ¬ r ) T T T T F F T T F T T T T F T T F F T F F T F F F T T T F F F T F T T T F F T F F T F F F F F T 1
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(b) Prove or provide a counterexample: A - ( B - C ) = ( A - B ) - C for any sets A,B, and C This is false. If A = B = C = { 1 } , then A - ( B - C ) = A - ∅ = A = { 1 } and ( A - B ) - C = ∅ - C = 3. (15 pts) (a) Find a solution to 63 x 3 (mod 132) We start wit hthhe extended Euclidean Algorithm: 132 = 2 · 63 + 6 63 = 10 · 6 + 3 6 = 2 · 3 + 0 Working backwards, we get 3 = 63 - 10 · 6 = 63 - 10
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This note was uploaded on 08/31/2009 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.

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final-sol - Math 55-2 Final Exam SOLUTIONS Rob Bayer August...

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