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Question: problem 1 consider three statements p q r statements 1...

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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 PA, (Q V R) and (P Л Q) V (PAR) This is one of the distributivity laws 2. Deduce the other distributivity law namely the logical equivalence between Pv(Q Λ R) and (PV(2) Λ(PVR) without using a truth table but instead using DeMorgans laws 3. Let P, Q, R and S be statements. Suppose that P is false and (R S) (P Л 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) (ab+2) is odd, then a and b are odd Problem 3. 1. Prove the following statement. For every a ER if a 2 4, then -+a< 0 2. Let aE R. Prove the following statement (a2 +az +a 0 for every zE R) if and only if (0< a<4) You may want to transform the expression za 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 A 1. There is a problem that has no solution 2. Every house in this city has at least 4 windows 3. For any IER, if x > 10 then 2-1 <1 4. For every Z, there exist numbers a Z and y E Z such that 22-2 +y2 Problem 5. (Chapter 4 Exercise 8). Suppose a is an integer. Prove the following statement if 52a, then 51a .

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