MATH130: Quiz 1

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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) T T F F F T F F T T F T T F T F F T T T

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).