Part II (10 points, 2 points per proof) For each of the following arguments construct a formal proof of validity by adding exactly three statements (do not use CP or RAA). A) 1. P→ Q 2. R v~Q 3. ~R~S B) 1. (P⚫Q) → R 2. (~R →~Q) → S ~Q) → S . P → S ~P C) D) 1. ~P v (Q⚫R) .. P→ Q 1. Pv~(Q ~R) • .. (P v ~Q) v R E) • 1. [Pv (Q ~R)] → S 2. Q 3. Q ~(P v ~R) .. (P v ~R) → S

Q: What is the equivalence of ( ¬ (( P → Q) ∧( Q → P))) ↔ ( Q → P)?

A: the equivalence of ( ¬ (( P → Q) ∧( Q → P))) ↔ ( Q → P) is to be computed below .

Q: Use 8 inference rules and 10 replacement rules (D.N., C.E., B.E., DeM., Dup., Assoc., Commu., Dist.,…

A: use 8 inference rules and 10 replacement rules  i) A≡~B∨C            ...iB≡D·~E…

Q: For each numbered line that is not a premise in each of the formal proofs that follow state the rule…

A: We prove the given argument by using rules of inference or replacement rules.

Q: For each numbered line that is not a premise in each of the formal proofs that follow state the rule…

A: Note: The line 8. W·X∨~W∨~X in the proof has a mistake. It should have been 8. W·X∨~W·~X. So, we…

Q: а. — (р @ q) + (р+q) b. (p ^ ¬q) → (p O r) → ¬p V q V ¬r с. (р — q)v(р r) —ръ(qvr) С.

A: To prove the following using equivalence law: a. ¬(p⊕q)⇔(p↔q) b. (p∧¬q)→(p⊕q)⇔¬p∨q∨¬r c.…

Q: d) (p→q)^(p →r) e) (p→q) v (q + r) f) (¬p →¬q) → (q + r)

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Q: 39. Construct a truth table for each of these compound propo- sitions. a) p→ (qVr) b) p→ (q→ r) c)…

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Q: W./.:// If P: p v(qar) and Q :(pvq)^(pvr) are J a logically Equivalent ( Prove that P Q)?

A: Two statement forms are called logically equivalent if, and only if, they have identical truth…

Q: 9. If g is true, which of the following statements' truth value can be determined? Give the truth…

A: q=True Part(i) q∧r=q AND r=True AND rCase 1: r=Trueq∧r=True AND True=TrueCase 2: r=Falseq∧r=False…

Q: R→T T :. P ¬Q) → (RAS) 1 2 3 4 5 7 8 9 STATEMENT R→ T (¬PV ¬Q) → (RAS) ¬RV¬S ¬¬PV¬Q) REASON…

A: I shall solve in details in next step

Q: 1) Let P, Q, and R be propositions. Consider the claim: (P → Q) ^ (R → Q) = (P V R) → Q. 1a) Use…

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Q: a) ((p v q) → ¬q) = q b) p→ (r→q) = (p^r) →q

A: Given  a) ¬((p ∨ q→¬q)≡q b) p→(r→q)≡(p∧r)→q To examine whether the relations a) ¬((p ∨ q→¬q)≡q and…

Q: 2. S → (T → U) U → -U (V → S) ^ (W → T) .. V → ~W

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Q: 4. The following are the proofing steps of an argument in formal reasoning. In proofing step 5, what…

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Q: 9. Prove valid 1. "G /NPK 2. "I (B > K) = "(K > G)] 3. ("B > T) & ("B Ə R) 4. (R & T) = (*N v G)

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Q: a) 1) The statement (p^q) vr⇒-pvr is not a tautology. A counter-example is p: T; q: F; r: T. p: T;…

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Q: b₂-d₂ (J (o- nu) va (4) A e) q→P d) p→q

A: D. If swimming at New Jersey shore is allowed, then sharks have not been spotted near the shore.

Q: Determine the line and the reason for the entry on line 6 given the following part of a formal proof…

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Q: 11. Let p and q be the propositions "Swimming at the New Jersey shore is allowed" and "Sharks have…

A: Disclaimer: Since, you have asked to solve only part G and H, these two parts have been solved only.…

Q: 5. For the given premises, determine a suitable conclusion so that the argument is valid. (a) p q, I…

A: Given

Q: - (р^ (q ~ г)) vr) and (p -> q) л —r с. d. pe (рлq) and p - q

A: First observe the given all laws and identities in the question carefully. And Just applying these…

Q: Assume X and Y are two statements. Which of the following is true? I. X^¬X is a contradiction. II.…

A: We have to solve given problem:

Q: Verify that the following statements are tautologies. 1) Transitive Law [(p → q)A(q → r)] → (p → r)…

A: 1)Transitive law[(p→q)∧(q→r)]→(p→r)2)Law of Excluded Middle p∨~p3)Law of Syllogism[p∧(p→q)]→q

Q: T 1 T. T F T 3 T F F 4 F T F T F T D7 F F F F F 8

A: Here we have to find only the truth table for the column 4. ~q means not q i.e. if q is true then ~q…

Q: 10. Write a valid conclusion for each argument A) ab B) a-b C) аль a

A: (a). For the given argument  a↔b~a--- Since the first argument is a is implies and implied by b and…

Q: sing truth tables. → P p→q) 9)

A:

Q: 5. Use abbreviated truth tables to show that each of the following arguments 1) Pv-Q, -R • S),…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Complete the truth table by filling in the required columns. 7) - р ^ р # дере }}} Ә) Т T F F P -…

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Q: The following set of statements establishes the validity of the given argument. State the rule of…

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Q: 1. Proof the following by deductive reasoning: a) (p → q) A (q → r) =p →r Transitive Law b) (p → r)…

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Q: Which of the following models are counter-models that shows that is not a logical consequence?…

A: We are given the following logical expression.        ∀x∃y P(x, y),  ∀x∃y P(x, y)→Q(x)⊭∀xQ(x)…

Q: III. From the given premise. Indicate the rules of inference used and the corresponding premises…

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Q: EXERCISES 1. Compute the truth values of the following, given that A, B, and C are true and X, Y,…

A: Since you have posted a multiple subparts question according to guidelines I will solve first(b,d,f)…