Question

Determine whether $\mathbf{\forall }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\leftrightarrow }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$and $\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\leftrightarrow }\mathbf{\forall }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}$are logically equivalent. Justify your answer.

Short answer

$\mathbf{\forall }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\leftrightarrow }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$and $\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\leftrightarrow }\mathbf{\forall }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}$are not equivalent logically.

Step-by-step solution

Definition of Logical equivalence

Logical equivalence is a relationship between two statements in propositional logic.

The two statements are not logically equivalent

LetP(x)be “xis divisible by2” and Q(x)be “xis divisible by4”

Thus, implies if xis divisible by2if and only if xis divisible by4which is false.

Also, in$\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\leftrightarrow }\mathbf{\forall }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ , for all x,xis divisible by2which is false and for all x,x is divisible by4is false $\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\leftrightarrow }\mathbf{\forall }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ is true.

Thus, $\mathbf{\forall }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\leftrightarrow }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$and $\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\leftrightarrow }\mathbf{\forall }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$are not logically equivalent.