By natural deduction, show the validity of Vx (P(x) V Q(x)), 3x ¬Q(x), Vx (R(x) → ¬P(x)) E 3a ¬R(x)
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- 40. Prove or disprove that .Prove that each argumentation below is valid using Natural Deduction: 1. Vx. (T(x) V S(x)) -T(a) : 3x. S(x) 2. Vx. (A(x) → (B(x) V C(x))) A(a) ^ ¬B(a) : C(a) 3. Vx. (A(x) → Vy. (B(y) → C(x, y))) А(m) л В(п) : C(m, n) 4. Vx. ((A(x) ^ B(x)) → C(x)) Vx. (A(x) → C(x)) : Vx.B(x) 5. 3x. A(x) → Vx. (B(x) → C(x)) 3x. D(x) → 3x. ¬C(x) 3x. (A(x) ^ D(x)) : -Vx. B(x)29 Use rules of inference to show that if Vx(P(x) v Q(x)), Vx(¬Q(x) v S(x)), Vx(R(x) → ¬S(x)), and 3X¬P(x) are true, then 3X¬R(x) is true.
- Use rules of inference to show Vx(R(x) v P(x)) is a valid consequence of the two premises: if Vx(P(x) V (Q(x)) and Vx(-P(x) A Q(x) → R(x))Which of the following logically negate the statement 'There is at least one plant that is not a flower? Select ALL that apply. OV (P(x)→ F(x)) 3x (P(x) ^¬F(x)) OVx-(P(x) A ¬F(x)) 0¬3x (P(x) ^ F(x)) 3x (P(x) V-F(x)) 03x (P(x) → ¬F(x)) OVE (-P(x) V F(x))¬p→(q→r) = q→(pvr) are logically equivalent?
- prove these deductions (a) x∩y=∅, (z⊂x), ∴(z∩y)=∅ (b) p⊂q, ∴(r∖q) = (r∖p)∖qWhich of the following statements is correct? a) P(X | Y) = P (X) if X and Y are independent. b) P (X | Y) = 1 / P (Y | X) c) P (X | Y) = P(X) / P(Y) d) None of the aboveUse rules of inference to show that if Vx(P(x) (Q(x) A S(x))) and Vx(P(x) A R(x)) are true, then Vx(R(x) A S(x)) is true. (Note: All the steps must be explained by reasons.)
- Let P(r) and Q(r) be predicates with one free variable, namely r. With explanation, are Vr (P(x) →¬Q(r)) and VxP(x) → -3xQ(x) logically equivalent?Let p denote 3x E Z such that P(x)" and q denote E R such that P(z" Give , simple definition for the predicate P(x) (but do not use "x 4 Z") so that q is true but n is false. Justify your answer.a) Use the rules of inference and laws of logic to prove the validity of the following argument. Provide the steps and reasons. [(-rvq)^(-s)^(-r→s)]= q