You can use your textbook, notes, class recordings, and any other written materials. But you must do your own work and must not discuss these problems with your classmates or anyone else. You are expected to do your own work without help from others. Please submit as a single Word doc. Your submission can include scanned images but they must be clear and legible. The due date and time of Tuesday, February 20 at 5pm Eastern time is a hard deadline; late submissions will not be accepted. Note that there are nine problems in total. Each of the problems carries roughly the same weight. 1. (a) Fill in the cells to complete the following truth table: (b) Based on your completed table, are ~(p vq) and ~p~q logically equivalent? Explain your answer. 2. Draw a truth table, and use it to determine whether the following argument form is valid. Include a sentence or two referring to your truth table that supports your answer: P4 ~r ~q . pVr 3. Find the value of each of the following expressions: (a) Zn=o(2" 1) (b) [T3=1(2k + 1) 4. Find the first six terms of the sequences defined by each of these recursive definitions: (@Qa=1;, a1=2; an=an1+3a,2forn>2 b)bo=1; bi=1; ba=by12by2forn=2 ()co=1;cn=2+c2_, forn>1 5. Use mathematical induction to prove that the sum of the first # odd positive integers is equal to n?; in other words, that 1 + 3+ 5+ -+ (2n 1) = n? forn > 1. 6. The arithmetic sequence ag, a,, a,, --* is defined recursively like this: _{b ifn =0 I =la,, +cifn>0 for constants b and c. Use mathematical induction to prove that the closed form is given by a, =b+ncforn=0. 7. Consider the following true statement S: n Z, if n is odd, then 2n is even. (a) Write the negation of statement S. (b) Write the contrapositive of statement S. (c) Write the converse of statement S. (d) Write the inverse of statement S. (e) With regard to the converse of statement S: If the converse is true then prove it; or if it is false then give a counterexample. 8. Prove this statement by contradiction: For all real numbers a and b, a + b = 2Vab. 9. Write each of the following in symbolic form and determine which two statements are logically equivalent (if any). Explain how you reached your conclusion: (a) If it flies but does not bark, it is a bird. (b) If it is not a bird, it either barks or does not fly. (c) It flies, and it does not bark and it is not a bird