Is the following proposition TRUE or false? Explain why. 1+1=3 if and only if 2+2=3. Write a proposition equivalent p v ¬ q using only p q ¬ and Λ . Write the contrapositive, converse, and inverse of the following “ if you try hard, then you will win” indicate clearly which is which. Suppose that Q(x)is ‘x+1=2x’ , where x is a real number. Give the truth value (TRUE or False) of each of following statement. Explain your answer. a. ∀xQ(x) b. ∃xQ(x) Consider the following theorem: If x is an odd integer, then x+2 is odd. Give a proof by contradiction of this theorem. 6. Decide if each of the following is TRUE or FALSE for all sets A,B and C. Justify your answers. a. A∪(B∩C)=(A∪B)∩(A∪C) b. If A=A∩C=B∩C,then A=B. 7 .Consider the function f(n)=2 ⌊n/2⌋ from Z to Z. Is this function one-to-one? Justify your answer. Is this function onto? Justify your answer. 8. Describe clearly ( in words or pseudocode ) an algorithm that takes a list of n integers a a_1, a_2…….., a_n and finds the number of integers in the list that are greater five. 9. What is the value of: -88 mod 3? 10. Use the Euclidean algorithm to find : gcd(390,72). 11- Prove or disprove: If p and q are primes ( p&q both >2), then p+q is composite. 12-If(n)= f(n-1) . f(n-2)+1 with f(0)= 1 and f(1)= 4, find f(5). 13. Explain the error in the following proof of a “theorem” Theorem: Every positive integer equals the next largest positive integer. 14. Suppose you wish to prove that the following is true forf all positive integers by using the principle of mathematical induction: 1+3+5+7+……..+(2n – 1)=n^2. write P(1) Write P(7) Write P(8) Write P(K) Write P(K+1) Use the principle of mathematical induction to prove that p(n) is true for all positive integers n.
Posted on: Sat, 15 Mar 2014 10:59:11 +0000
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