1. (18 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. _If 2 + 2 = 5 and 1+2 = 3, then 10 +5 = 14 or 3 + 2 = 5. 3 + 4 = 2 if and only if 2 + 3 = 5. ("7 >6" A "2 * 3 = 6) v 3>5" pvF=p, where p is a proposition and T is tautology. 2 V p A r = o A q V r 23 x P(x) is equivalent to 3 x P(x). _3 x 3 y P(x, y) is equivalent to 3 y 3 x P(x,y). 3 y Vx P(x, y) is not equivalent to V x 3 y P(x,y). | + 2 = 3" WOR 3 + 3 = 6" 1 (2 ] ( ) = mp A 14 A mr. pVqA F= F, where F is contradiction. A proposition is also a propositional function. 2. (5 points) Use the truth table to prove or disprove (p+q) = p1-9. 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). V x 3 y P(x,y). (b). Hy V P(x,y). 4. (6 points) Let P be the statement If n 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. (6 points) Let a @ b = max{a,b} = a if b