Problem 1. Consider three statements P , Q, R statements.
1. Using truth tables, recall the proof of the logical equivalence between
P ∧ (Q ∨ R) and (P ∧ Q) ∨ (P ∧ R). This is one of the distributivity laws.
2. Deduce the other distributivity law namely the logical equivalence between
P ∨ (Q ∧ R) and (P ∨ Q) ∧ (P ∨ R).
without using a truth table but instead using DeMorgan’s laws.
3. Let P , Q, R and S be statements. Suppose that
P is false and (R ⇒ S) ⇔ (P ∧ Q) is true. Find the truth values of R and S.
Problem 2. Prove the following statement:
Let a and b be integers. If (a − 2)2(ab + 2) is odd, then a and b are odd.
Problem 3. 1. Prove the following statement. For every a ∈ R.
if a ≥ 4, then − 4 + a ≤ 0.
2. Let a ∈ R. Prove the following statement:
(x2 + ax + a > 0 for every x ∈ R) if and only if (0 < a < 4).
You may want to transform the expression x2 + ax + a by completing the square.
Problem 4. Write the negation of the following statements. Avoid using ”there is no”, ”is not” as much as possible. You may use the sign =.
1. There is a problem that has no solution.
2. Every house in this city has at least 4 windows.
3. For any x ∈ R, if x > 10 then 2−x < 1.
4. For every z ∈ Z, there exist numbers x ∈ Z and y ∈ Z such that z2 = x2 + y2.
Problem 5. (Chapter 4 Exercise 8). Suppose a is an integer. Prove the following statement:
if 5|2a, then 5|a .