1. Proof the following by deductive reasoning: a) (p → q) A (q → r) =p →r Transitive Law b) (p → r) V (q → s) = (p ^ q) → (r V s) Constructive Dilemmas (CD)

Problem 13. What are L and D for this matrix A? What is U in A = LU and what is the new U in A = LDU? A = = [2 4 8 039 007 .

7. Negate the following statements: а. (Vx € P) (х +4 0

Question 4. The following are logical laws shown in our lecture. Find their dual versions if exist. Implication law Idempotent law Соmmutative law A → B = ¬AV B AA A = A АЛВ 3D ВЛА (АЛ В) ЛС 3D Ал (ВЛС) Ал (AV B) А A A (B V C) Associative law Absorption law = (A ^ B) V (A ^ C) Distributive law Double Negation law De Morgan's law Excluded middle law A ¬(A ^ B) = ¬A V¬B AA¬A = F АЛТ 3D А Identity law Domination law AAF = F

Which of the following statements is true when A is true and B is false? 1 ~A = (A ɔ B) 2 A = (~B ɔ ~A) 3 A = ~(A v B) 4 ~A = (A v B)

7. Use all of the premises to determine a alid conclusion for the following argument. ~m Vt t → -d e Vg e → d ..?

State the rule of inference by which its conclusion follows from its premise or premises. (H ɔ (I = J)) v (E v F) - (H ɔ (I = J)) .: (E v F) A Modus Ponens B Disjunctive Syllogism Modus Tollens Constructive Dilemma

2. Let p, q, and r be as in Exercise 1. Translate the follow- ing into English sentences. (a) (p ^ q) → r (c) p (q vr) (e) (pvq) ^r (b) (pr)→ 9 (d) (p → (q vr))