Question

Establish these logical equivalences, where x does not occur as a free variable in A. Assume that the domain is nonempty.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{(}\mathbf{\forall }\mathbf{xP}\mathbf{}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{\wedge }\mathbf{A}\mathbf{}\mathbf{\equiv }\mathbf{\forall }\mathbf{x}\mathbf{(}\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{A}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{(}\mathbf{\exists }\mathbf{xP}\mathbf{}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{}\mathbf{\wedge }\mathbf{A}\mathbf{\equiv }\mathbf{\exists }\mathbf{x}\mathbf{(}\mathbf{P}\mathbf{}\mathbf{(}\mathbf{x}\mathbf{)}\mathbf{\wedge }\mathbf{A}\mathbf{)} \end{gathered}$$

Short answer

$\mathrm{a})\mathbf{}\mathbf{\forall }\mathbf{xPx}\mathbf{\wedge }\mathbf{A}\mathbf{}\mathbf{\equiv }\mathbf{\forall }\mathbf{x}\mathbf{(}\mathbf{Px}\mathbf{\wedge }\mathbf{A}\mathbf{)}$the two propositions have the same truth value and thus they are logically equivalent.

$\mathrm{b})\mathbf{}\mathbf{(}\mathbf{\exists }\mathbf{xPx}\mathbf{)}\mathbf{}\mathbf{\wedge }\mathbf{A}\mathbf{\equiv }\mathbf{\exists }\mathbf{x}\mathbf{(}\mathbf{Px}\mathbf{\wedge }\mathbf{A}\mathbf{)}$the two propositions have the same truth value and thus they are logically equivalent.

Step-by-step solution

(∀xP (x))∧A ≡∀x(P(x)∧A)

a) If $\mathbf{(}\mathbf{\forall }\mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{A}$is true, then A is true and for all values y we have thatP(y) is true. Then $\mathbf{P}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{\wedge }\mathbf{A}$is true for all values of y, which means that $\mathbf{(}\mathbf{\forall }\mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{A}$ is true.

If $\mathbf{(}\mathbf{\forall }\mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{A}$is false, then A is false or there exists a value ysuch that P(y) is false. Then $\mathbf{P}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{\wedge }\mathbf{A}$is false, which means that $\mathbf{(}\mathbf{\forall }\mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{A}$ is false.

Thus the two expressions always have the same truth value and thus they are logically equivalent.

(∃xP (x)) ∧A≡∃x(P (x)∧A)

b) If $(\exists \mathrm{xP}(\mathrm{x}))\wedge \mathrm{A}$is true, then A is true and there exists a value for which P(y) is true. Then P (y) v A is true, which means that $\exists \mathrm{x}(\mathrm{P}(\mathrm{x})\vee \mathrm{A})$is true.

If $(\exists \mathrm{xP}(\mathrm{x}\left)\right)\wedge \mathrm{A}$is false, then A is false or for all values we then have that P(y) is false. Then $\mathrm{P}\left(\mathrm{y}\right)\wedge \mathrm{A}$ is false for every value of, which means that $\exists \mathrm{x}\left(\mathrm{P}\right(\mathrm{x})\wedge \mathrm{A})$ is also false.

Thus the two expressions always have the same truth value and thus they are logically equivalent.