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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.

a2

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 2x < 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 .

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