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# e3 - 2n=n(n 1(a What are i(2 points p(1 ii(2 points p(2...

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CSE 260 Exam3, November 14, 2005 Name: This exam has 3 pages and 10 problems totaling 100 points. This is closed book and closed notes. A student can bring in a 8 1/2x11 sheet of notes. 1. (5 points) What is the nth terms of the following sequence. 5,10,20,40,80,160,320,. . . 2. (5 points) Find the value of Σ 100 k =10 k Given Σ n k =1 k = n ( n + 1) / 2 3. (2 points) Evaluate the following: -77 mod 6 4. Answer the following true/false (a) (2 points) 29 8(mod 7) (b) (2 points) 31 -14(mod 3) 5. (10 points) Which positive integers between 20 and 30 are relatively prime to 20? 6. Indicate if the following sets are countable. if the set is countable, give the one-to-one correspondence between Z + and the set. (a) (5 points) Set of all odd positive integers. 1

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(b) (5 points) Set of all odd integers. (c) (5 points) Set of all reals between 0 and 1. 7. Let p(n) be the proposition p(n): 2+4+6+8+ . . .+2n=n(n+1)
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Unformatted text preview: . .+2n=n(n+1) (a) What are i. (2 points) p(1): ii. (2 points) p(2): iii. (2 points) p(n+1): (b) (10 points) Prove p(n) ∀ n by mathematical induction. 8. (10 points) p ( n ) : n 2 + 5 n + 1 is odd Prove p(n) ∀ n ≥ 1 by mathematical induction. 2 9. Following relation R is represented by a directed graph (digraph). A B C D (a) (5 points) Write the relation R as a set of elements of R × R . (b) (12 points) Circle the appropriate properties below for the relation. refexive irrefexive symmetric antisymmetric asymmetric transi-tive (c) Find the powers R 2 , R 3 (5 points) R 2 = (5 points) R 3 = 10. Following relation is de±ned on the set A = { 1 , 2 , 3 , 4 } R = { ( a, b ) ²A × A | a ≡ b (mod 2) } (a) (5 points) Give the relation R as a set. (b) (1 points) Is it an equivalence relation? 3...
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e3 - 2n=n(n 1(a What are i(2 points p(1 ii(2 points p(2...

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