Ministry of Higher Education Kingdom of Saudi Arabia CSTS SEU, KSA Discrete Mathematics (Math 150) Level III, Assignment 1 (2017) 1. State whether the following statements are true or false: [6] (a) 12+4=7-16 is a proposition. (a) (b) The AND operation of two propositions p and q denoted by p q is true if both propositions are not true. (b) (c) In Boolean algebra, x = x. (c) (d) Boolean algebra simplifies logic circuits. (d) (e) The nested quantifier with real numbers x, y, xyP (x, y), where P (x, y) : x + y = 0, is true. (e) (f) If Q(x, y) : x + y = x y with domain consists of all integers, then xQ(x, 2) is False. (f) Page 1 of 3 Due Date: 22.10.2017 Math 150 Department of Mathematics 2. Select one of the alternatives from the following questions as your answer. (a) The inverse of the proposition q p is A. p q B. p q C. q p D. p q (b) The symbolization for a conjunction of two propositions p and q is A. p q B. p q C. p q D. p q (c) A boolean function may be transformed into a A. logical expression. B. map. C. matrix. D. none of the above. (d) The degree of the Boolean function given by f (x, y, z, w) = xy + yz + zw is A. 2 B. 4 C. 3 D. 1 (e) Translate xy(x < y) in English, considering domain as real number for both the variable. A. For every real number y there exists a real number x such that x is less than y. B. For some real number x there exists a real number y such that x is less than y. C. For all real number x there exists a real number y such that x is less than y. D. For each and every real number x and y such that x is less than y. Page 2 of 3 Due Date: 22.10.2017 [6] Math 150 Department of Mathematics (f) Express, The difference of a real number and itself is zero using required operators. A. x(x x! = 0) B. xy(x y = 0) C. x(x x = 0) D. x(x x = 0) 3. Construct the truth table for the compound proposition [3] (p q) (p q). 4. Show that the conditional statement (p q) (p q) is a tautology. [3] 5. Use a table to express the values of the boolean function given by [3] f (x, y) = x y + xy. 6. Construct a combinatorial circuit using inverter, OR gates and AND gates that produces the output pq + pq, from input bits p and q. [3] 7. Express the following sentences in predicate quantifiers notations: [3] (a) Every comedian is funny. (b) There is a comedian who is not funny. 8. Use predicate quantifiers, logical connectives and mathematical operators to express the statement that the average of two positive real numbers is always positive. Page 3 of 3 End of Assignment. [3]