1. Create the entire truth table for this statement. p^ (qVr) 2. (10 POINTS) Create the...

 

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1. Create the entire truth table for this statement. p^ (qVr) 2. (10 POINTS) Create the entire truth table for this statement. (p^q) V (p^r) 3. (2 POINTS) Refer to your answers to Problems 1 and 2 to answer this question: Is p^ (qVr) = (p^q) V (p^r)? Explain how you reached your conclusion. [You do not need to create a truth table for this problem.] 4. (3 POINTS) The domain for this question is the set of all integers. Consider this wff: (3x) (y) [r² + y² = 5] If this has truth value T explain why, and if it has truth value F explain why. 5. (5 POINTS) Give a direct proof [showing all steps] of the following: If m is an even integer and n is an odd integer then m² +n²+1 is an even integer. 6. (5 POINTS) Use the method of contradiction to prove the following [all details and steps must be written out]: If 11n + 6 is odd then n is odd. 7. (10 POINTS) Prove If n is an integer and n² + 13 is odd then n is even. State the type of proof (direct, contraposition, or contradiction) you're using and show all steps of the proof. 8. Use a = 260, b= 120 for the parts of this problem. (a) (8 POINTS) Find the prime factorization of a. (b) (8 POINTS) Find the prime factorization of b. (c) (3 POINTS) Use your answers to parts 8a and 8b to find the prime factorizations of gcd(a, b) and lcm(a, b). 9. Use m = 345 and n = 240 for the parts of this problem. All of the work in each part must be shown. (a) (10 POINTS) Use Euclid's Algorithm to find gcd (m, n). (b) (10 POINTS) Use your answer to part 9a to write gcd (m, n) as a linear combination of m and n. 10. (6 POINTS) Consider the sequence that has the following recursive def- inition: a (1)=2, a (2) = 1, a (n) = 5a (n-1) +6a (n-2) Vn ≥ 3 Find a (3), a (4), and a (5). 11. The recursion relationship in Problem 10 is a linear constant-coefficient relationship. As discussed in class assume a (n) = Br". (a) (5 POINTS) Show the work that leads to the equation that r must satisfy for this sequence. [Hint: you should end up with a quadratic equation.] (b) (5 POINTS) Solve the equation from part 11a to find the two values of r. 12. Consider the following. n P(n): Σ k=2 1 k (k-1) n-1 n Vn 22 (a) (2 POINTS) Show that the anchor case P(2) is true. (b) (4 POINTS) Write out the induction hypothesis. (c) (4 POINTS) Write out the expression you need to prove. (d) (12 POINTS) Use the First Principle of Mathematical Induction to prove the result. All steps must be shown. (e) (3 POINTS) Evaluate P(200) (leave your answer as a fraction). 1. Create the entire truth table for this statement. p^ (qVr) 2. (10 POINTS) Create the entire truth table for this statement. (p^q) V (p^r) 3. (2 POINTS) Refer to your answers to Problems 1 and 2 to answer this question: Is p^ (qVr) = (p^q) V (p^r)? Explain how you reached your conclusion. [You do not need to create a truth table for this problem.] 4. (3 POINTS) The domain for this question is the set of all integers. Consider this wff: (3x) (y) [r² + y² = 5] If this has truth value T explain why, and if it has truth value F explain why. 5. (5 POINTS) Give a direct proof [showing all steps] of the following: If m is an even integer and n is an odd integer then m² +n²+1 is an even integer. 6. (5 POINTS) Use the method of contradiction to prove the following [all details and steps must be written out]: If 11n + 6 is odd then n is odd. 7. (10 POINTS) Prove If n is an integer and n² + 13 is odd then n is even. State the type of proof (direct, contraposition, or contradiction) you're using and show all steps of the proof. 8. Use a = 260, b= 120 for the parts of this problem. (a) (8 POINTS) Find the prime factorization of a. (b) (8 POINTS) Find the prime factorization of b. (c) (3 POINTS) Use your answers to parts 8a and 8b to find the prime factorizations of gcd(a, b) and lcm(a, b). 9. Use m = 345 and n = 240 for the parts of this problem. All of the work in each part must be shown. (a) (10 POINTS) Use Euclid's Algorithm to find gcd (m, n). (b) (10 POINTS) Use your answer to part 9a to write gcd (m, n) as a linear combination of m and n. 10. (6 POINTS) Consider the sequence that has the following recursive def- inition: a (1)=2, a (2) = 1, a (n) = 5a (n-1) +6a (n-2) Vn ≥ 3 Find a (3), a (4), and a (5). 11. The recursion relationship in Problem 10 is a linear constant-coefficient relationship. As discussed in class assume a (n) = Br". (a) (5 POINTS) Show the work that leads to the equation that r must satisfy for this sequence. [Hint: you should end up with a quadratic equation.] (b) (5 POINTS) Solve the equation from part 11a to find the two values of r. 12. Consider the following. n P(n): Σ k=2 1 k (k-1) n-1 n Vn 22 (a) (2 POINTS) Show that the anchor case P(2) is true. (b) (4 POINTS) Write out the induction hypothesis. (c) (4 POINTS) Write out the expression you need to prove. (d) (12 POINTS) Use the First Principle of Mathematical Induction to prove the result. All steps must be shown. (e) (3 POINTS) Evaluate P(200) (leave your answer as a fraction).