1. (12 points) Determine if the following propositions are TRUE or FALSE. Note that p, q, r are propositions. P(x) and P(x,y) are predicates. (a). _______ If 2 + 2 = 5 and 1 + 2 = 3, then 10 + 5 = 14 or 3 + 2 = 5. (b). _______ 3 + 4 = 2 if and only if 2 + 3 = 5. (c). _______ (7 > 6 2 * 3 = 6) 3>5 (d). _______ F , where is a proposition and T is tautology. (e). _______ (f). _______ () is equivalent to (). (g). _______ y (, y) is equivalent to y (,y). (h). _______ y (, y) is not equivalent to y (,y). (i). _______ 1 + 2 = 3 XOR 3 + 3 = 6 (j). _______ ( ) p r (k). _______ , where is contradiction. (l). _______ A proposition is also a propositional function. 2. (5 points) Use the truth table to prove or disprove ( ) p .

3. (5 points) Suppose P(x,y) means x * y = y * x, where x and y are real numbers. Determine the truth value for each of the following statements. (a). y(,y). (b). y (,y).

4. (6 points) Let P be the statement If 2 is an odd integer, then n is odd integer. (a). What is the contraposition of P? (b). Prove P by contraposition.

5. (4 points) Let S(x,y) be the statement student x likes soft drink y , where the domain for x consists of all students at your school and the domain for y consists of all soft drinks. Use quantifiers to express Not every student likes tea and soda.

6. (8 points) Let a @ b = max{a, b} = a if b a, otherwise a @ b = max{a, b} = b. Give a proof by cases that for all real numbers a, b, c (a @ b ) @ c = a @ (b @ c)

7. (10 points) Using truth table prove that- pq (pq)(pq)