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## Description

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.} ^{Dedu}^{c}^{e} ^{the} ^{other} ^{distributivi}^{t}^{y} ^{l}^{a}^{w} ^{namely} ^{the} ^{logical} ^{equi}^{v}^{alence} ^{b}^{e}^{t}^{w}^{een}

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.

_{a}^{2}

^{if} ^{a} ^{≥} ^{4,} ^{then }^{−} _{4 } ^{+} ^{a} ^{≤} ^{0.}

2. Let a ∈ R. Prove the following statement:

(x^{2 } + ax + a > 0 for every x ∈ R) if and only if (0 < a < 4).

You may want to transform the expression x^{2} + 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 z^{2 } = x^{2} + y^{2}.

Problem 5. (Chapter 4 Exercise 8). Suppose a is an integer. Prove the following statement:

if 5|2a, then 5|a .

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