MATH130: Quiz 1

You may use a calculator on this quiz. If you get stuck on a question, move on to the next. If you have time left over at the end of the quiz, check your work.

  1. Shown below is the truth table for the statement (¬p) ∨ (¬q).
    a) (15 pts) Construct the truth table for ¬(p ∨ q).

    p q ¬p ¬q (¬p) ∨ (¬q)
    TTF FF
    TFF TT
    FTT FT
    FFT TT






    b) (10 pts) Is ¬(p ∨ q) equivalent to (¬p) ∨ (¬q)? Explain.





  2. (25 pts) Suppose P(x) represents the propositional function x² > x+1 and the domain of discourse D is all positive numbers. Is ∀x P(x) a true or false statement? Explain.






  3. If A = {a, b, c, d} and B = {c, d, e, f}
    a) (15 pts) What is A ∩ B?



    b) (10 pts) What is A - B?



  4. To prove the propositional function S(n): 1 + 2 + 3 + ... + (n-1) + n = n*(n+1)/2 by induction, we first proved S(1).
    a) (10 pts) What is S(1)?




    We next assumed that 1 + 2 + 3 + ... + (k-1) + k = k*(k+1)/2 (statement S(k)) and used this assumption to prove S(k+1).
    b) (15 pts) What is S(k+1)?




Bonus: (5 pts) As described in the problem above, assume S(k) is true and use this to prove S(k+1).