Ministry of Higher Education Kingdom of Saudi Arabia CSTS SEU, KSA Discrete Mathematics (Math 150) Level III, Assignment 1 (2015) 1. State whether the following statements are true or false: [9] (a) y (x y) is a tautology. (a) (b) The compound propositions (p q) and p q are not equivalent. (b) (c) The negation of conjunction of two propositions is equivalent to the disjunction of the negation of those propositions. (c) (d) In Boolean algebra, 1 + x = x. (d) (e) The product of sums is basically the ORing of ANDed terms. (e) (f) The dual of (x + y).z is x.y + z. (f) (g) The universal quantication xP (x) is false only when P (x) is false for each value of x in the domain. (g) (h) The quantied statement x, y N, x + y > x x + y > y is true. (h) (i) Negation of x(p(x) q(x)) is xp(x) q(x). (i) Page 1 of 4 Please go on to the next page. . . Math 150 Department of Mathematics 2. Select one of the alternatives from the following questions as your answer. (a) The compound proposition p q is equivalent to A. p q B. (p q) C. p q D. None (b) The converse statement of the compound proposition q p is A. p q B. q p C. p q D. None (c) The compound proposition p (p q) is equivalent to A. p B. q C. p q D. p q (d) Which of A. B. C. D. the following expression is in the sum-of-products form? (x + y).(z + w) x.y + z.w (x)y(zw) None of above (e) The values of x, y, z and w that makes the sum term x + y + z + w equal to zero is A. x = 1, y = 0, z = 0, w = 0 B. x = 0, y = 1, z = 0, w = 0 C. x = 1, y = 0, z = 1, w = 1 D. x = 1, y = 0, z = 1, w = 0 (f) The resulting Boolean expression for the following circuit is Page 2 of 4 Please go on to the next page. . . [9] Math 150 Department of Mathematics A. B. C. D. (C(A + B)D) + E [(C(A + B)D)]E ABCDE C(A + B)DE (g) The multiplicative inverse law for the nonzero rational numbers is given by A. xy (xy = 1) B. xy (xy = 1) C. xy (xy = 1) D. all of the above. (h) The negation A. x, B. x, C. x, D. x, of x(p(x) q(x)) is p(x) q(x) p(x) q(x) p(x) q(x) p(x) q(x) (i) When multiple quantiers dier, then the meaning of a predicate logic sentence A. is determined by arithmetic operators B. depends on order C. is ambiguous D. is independent of order 3. Construct the truth table of compound proposition (p q) (p r). [2] 4. Find the contrapositive and inverse statements of the statement \"If you do every exercise in this book then you are a good student\" [2] 5. Find the values of the boolean function f (x, y, z) = xy z + xyz + xy. z [2] 6. Determine the output of the following combinatorial circuit: [2] 7. Prove that n2 2 is not divisible by 5. [2] Page 3 of 4 Please go on to the next page. . . Math 150 Department of Mathematics 8. Let the domain for x be the set of integers. p(x) : x is even. q(x) : x is a prime number. r(x) : 5 divides x. [2] Write the following symbols as quantied statements. 1. x, (p(x) q(x)). 2. x(p(x) q(x)) r(x). Page 4 of 4 End of Assignment