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Let P(x) and Q(x) be arbitrary predicates. Which of the following statements is always TRUE?

A

((∀x(P(x)∨Q(x))))⟹((∀xP(x))∨(∀xQ(x)))

B

(∀x(P(x)⟹Q(x)))⟹((∀xP(x))⟹(∀xQ(x)))

C

(∀x(P(x))⟹∀x(Q(x)))⟹(∀x(P(x)⟹Q(x)))

D

(∀x(P(x))⇔(∀x(Q(x))))⟹(∀x(P(x)⇔Q(x)))

asked in Propositional and first order logic by gate

2 Answers

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(∀x(P(x)⟹Q(x)))⟹((∀xP(x))⟹(∀xQ(x)))

answered by gate
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A    ((∀x(P(x)∨Q(x))))⟹((∀xP(x))∨(∀xQ(x)))
answered by Mansi

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