Instruction: Work out (give only the process and answers, use clear hand writing). 1. Determine whether each of the following compound propositions is a tautology, a contradiction or neither. (2point each) a). ((p q) ^ (qr)) (pr) b). [p ^ (q V r)] [(p ^ q) v (p v r)] 2 Determine the truth values the following statements. (1point each) a). (3x E R) (3y E R) (x + y + 3 = 8). b). (VxER) (Jy R) (x + y = 5). 3. validity. a. If he studies medicine, he will get a good job. If he gets a good job, he will get a goo d wage. He did not get a good wage. Therefore, he did not study medicine. (2 point) 4. How many elements a. b. c. 255 proper subsets? No Check Let a the does A contain if it has: (1point each) 64 subsets subset? proper U = {2, 3, 6, 8, 9, 11, 13, 15). factor of 66} 5. A = {x | x is positive prime B = {x EU | x is composite number} and C = {x E Ux-5 EU}. Then find each of the following: (1point each) a). An B b). (AUB) nC c). An B' d). A AB e). A'n B'n c' 6. 10 0 0 0 OHIN 2 00 W|T 3 A O O CHINO 0 D)-( = 1 0 0 1 00 -2 0 1 7. Given that 2 1 0 A-2 0 0 1 0 0 1/ 8. For which numbers a and b does the following system have (2point) A). a unique solution B). no solution C). Infinitely many solution x - 2y + 3z = 4 2x - 3y + az = 5 3x - 4y + 5z = b 9. Solve the following system using Gauss elimination method (2point) x-y+z=5 3x + y -z = 4 -x+y=z= -3 aij = then find A. (2point) 10. Find all values of a, b and c for which matrix A is symmetric, where (1 point) [2 a 2b + 2c 2a+b+c 3 5 Lo a+c 11. Construct a 3x2 matrix A = (a) such that 2 0 1 0 = 513, what is det(A)? (2point) # 72 +9 -2 (j-2i, for i