In the following, prove some alternative rules of inference. a) Strengthening  { (P → Q ) } ⊢ ((P  ∧ R ) → Q ) [5 lines]   b) Weakening  { (P → Q ) } ⊢ (P → (Q  ∨ R )) [6 lines]   c) Double Negation Introduction { P  } ⊢ ~~P [5 lines]

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: Part 2. Prove that the following are equivalent A. by Constructing a truth table 1. [p V (~ p ^q)] →…

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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: Determine which of the following statements are true, and which are false. Explain! (a) {a} ⊆ {a, b,…

A: Given that: (a)  a⊆{a, b, c}(b)  {a}⊆{{a,b},c}(c){a}⊆P{{a}, b, c}

Q: Determine if each form is a tautology, a contradiction, or a contingency: a)(p^-q)^(-p vq)…

A: Proposition: A proposition is a statement that is either true or false.A compound proposition can be…

Q: qV[~(q^r) ~q]

A:

Q: Prove: PAQ -QAP using inference rules

A: To Prove- P∧Q - Q∧P by using Inference Rules.     Rules Used-  Simplification: P∧Q∴P Conjunction:…

Q: (3rt)(は)(キ)… 。 Sn-? S- ?

A: We will find out the required value.

Q: Write above five statements into Mathematical logic b) Show that the conditional statement [-p^ (p V…

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Q: Construct truth tables for the following and identify any Tautologies - (p A q)→(-p V q) I (p-> q )…

A: We will find out the required value.

Q: B. by using Rules of Replacement 1. [p V (~ p ^q)] → (p A q) 2. {~ (pvq) v [(~ p^ q) v ~q]} +→ ~…

A:

Q: When "→" is understood as the material conditional, P(p → q) is equal to Multiple Choice   P(p) +…

A: When "→" is understood as the material conditional, P(p → q) is equal to

Q: Which of the following statements is a tautology? A. (p V ~ q) → (q ^ ~ r) B. - [r → (q A r)] C. q…

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Q: А. а —> (bлс) (ьле) —d ..a»d jak В. :(j^k)V(I^m)

A:

Q: Determine the truth value of the statement (~r^ p) Aq using the following conditions. a) p is false,…

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Q: 2. By building tables check which statements are tautologies. a) (p ^ q) => (q v p) b) p => - p c)…

A: As per our guidelines, we are supposed to solve only first three subparts. Kindly repost other…

Q: (p^ (~(~pV q))) v (pAq) = p.

A: Few notation for the problem. p∧q : p and q [ This will be true if both p and q are true] p∨q : p or…

Q: Are all three answers correct

A: Given sets of vectors are,(a) (b) (c) The objective is to determine whether the set is a basis of…

Q: II. Prove/Disprove: 1. (¬A → → ¬B) + (¬C A A) = ¬ (A V –¬B) + (В — С)

A: We will check whether the given statement is equivalent or not.

Q: Which is the contrapositive of A==> B? (a) B ==> A (b) -,B ==> A (c) -,B ==> -,A (d) -,A…

A: Which is the contrapositive of A==> B?

Q: Question 2 Is (p → q) why not? (-q → -p) a tautology? Why or

A: Note:- As per our guidelines, we can answer the First part and its sub parts only if any particular…

Q: 1. ~(AV B) — с 2. ~A 3. ~С Show that (1) – (3) entails B

A: We have to show that (1)-(3) entails B using the rule of inference.

Q: The sentences (A&B) → ~C and ~C V ~(B & A) arenot equivalent. Select one: O True O False age

A: True

Q: sing truth tables. → P p→q) 9)

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Q: ((,ô^‚d) v (o^a)) = ((d←Ô) v (Ô←d))

A:

Q: i. ii. iii. Determine whether the formulae below are tautologies (A → ((¬B) V C)) → ((¬A) → B) (A →…

A: The truth table is defined as a tabular representation of the expression of all the combinations of…

Q: 2. Write negations of the following conditional statements: - (p-→q)= p^~q (a) VneZ,ifn is divisible…

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Q: e) [Reminder: replacement rules, such as 1. (~Av-(AvB)) (AvB) 2-AD-(BvC) 3.-~(BvC)• (~Bv~A) 4.…

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Q: discrete math Use truth tables to show that ~ (p →→q) and p^~ q are logically equivalent

A: Given that the statement ~p→q  and  p∧~q  .

Q: e) (pvq) ^¬

A:

Q: Determine what rule of replacement was used: (H כ) י (F כ E) .. (E ɔ F) . ( ~~H ɔ ~ G) (A) Material…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.