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CSTS-SEU-KSA Discrete mathematics- Math150 Assignment 1 2nd semester 2016-2017 ------------------------------------------------------------------------------------------Section I Determine whether the statement is true or false. (1 mark for each) 1) x + y = z is a proposition. 2) The compound proposition p ∧ p is a tautology. 3) There are 256 different Boolean functions of degree 3. 4) If 5) ( , then ̅̅̅̅̅̅̅̅̅̅̅̅ ( )= ) ̅̅ ̅ The argument in propositional logic is valid if the premises imply the conclusion. 6) The expression “ (¬ p∧ q) ∨ r” is in disjunctive normal form (DNF) Q 1 f 2 f 3 T 4 F 1 5 T 6 T CSTS-SEU-KSA Section II Choose the correct answer 1) The converse of the statement q →r (1 mark for each) is a) q →r b) ¬ q →¬r c) r →q d) ¬ r →q 2) If P: It is raining , and Q: I will study discrete mathematics, then the statement” if It is raining, then I will study discrete mathematics” represents P∨q P∧q Pq Pq a) b) c) d) 3) Boolean expression for the Boolean function F(x, y) which defined by the table below is 0 0 1 1 a) b) c) d) ̅̅ ̅ ̅ ̅̅̅ F( , ) 1 1 1 0 0 1 0 1 ̅ 2 CSTS-SEU-KSA 4) The representation of the sentence "Every student in this university took a course in mathematics," in predicate logic, is ----------a) xMx b) xMx c) x ¬ Mx d) x ¬ Mx 5) Assume that p ↔q is True, then a) b) c) d) Both of p and q must be same , otherwise false Both of p and q must be true , otherwise false No one of p and q must be true , otherwise false Both of p and q must be false , otherwise false 6) Let Mx,y denoted by “ ”,where , then -----------is True. a) xyMx b) y xMx c) y x Mx d) xy ¬ Mx 1 C 2 C 3 A 4 A 5 A 3 6 B CSTS-SEU-KSA Section III Solve the following questions (3 marks for each) 1) Construct a truth table for the proposition (¬ p → q) ∧ (q → r) → r. P T T F F F T F T q T F T F T F F T r F T T T F F F T ¬p→q T T T F T T F T q→r F T T T F T T T (¬p → q) ∧ (q → r) F T T F F T F T (¬ p → q) ∧ (q → r) → (r) T T T T T F T T 2) Show that p ↓q is logically equivalent to ¬ p∨ q) by using a truth table p T T F F q T F T F p ↓q F F F T ¬ p∨ q) F F F T 4 CSTS-SEU-KSA 3) Find the output of the following circuit P q r T T F F F T F T T F T F T F F T F T T T F F F T r T F F F T T T F r ∨ q r ∨ q) ∧ P T F T F T T T T T F F F F T F T 4) Find the sum-of-products expansions of the Boolean function F ( , ) = , by using a truth table. x 0 1 1 0 y 1 0 1 0 o o 1 o F( , )= 5 CSTS-SEU-KSA 5) What is the truth value of ∃xP(x) and xP(x), where P(x) is the statement “x2 > 5” and the domain is {1, 2, 3}? ∃xP(x)= P(1) ∨ P(2) ∨ P(3)=T xP(x)= P(1) ∧ P(2) ∧ P(3)=F 6 CSTS-SEU-KSA 6) Find the Disjunctive Normal Form (DNF) of ((p∨q) ∨ (s∨r) ) → ¬ (p∨q) SOLUTION (¬p∧¬q) ∨ ¬((p∨q) ∨ (s∨r)) Construct the truth table for the proposition. Then an equivalent proposition is the disjunction with n disjoints (where n is the number of rows for which the formula evaluates to T). Note that, we used this idea in chapter 12. p q r s ¬p∧¬q p∨q s∨r (p∨q) ∨ (s∨r) ¬((p∨q) ∨ (s∨r)) T T F F F T F T T T F F F T F T T F T F T F F T T F T F T F F T F T T T F F F T F T T T F F F T F T T T F F F T T F F F T T T F f f f T f f T f f f f T f f T f t t t F t t F t t t t F t t F t F t t t F F F t t t t t t t t t T T T T T T f T T T T T T T T T f f f f f f T f f f f f f f f f ((p∨q) ∨ (s∨r) ) → ¬ (p∨q) F F F T F F T F F F F T F F T F (¬p∧¬q∧r∧s) ∨ (¬p∧¬q∧¬r∧¬s) ∨ (¬p∧¬q∧r∧¬s) ∨ (¬p∧¬q∧¬r∧s) 7