Question

Show that $\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$ and $\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{\exists }\mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$ are logically equivalent

Short answer

$\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$and$\mathbf{\exists }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\vee }\mathbf{\exists }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ are 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 logically equivalent

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

Thus,$\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$states there existsxsuch thatxis divisible by2orxis divisible by4which is true.

Also,$\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$states that, there exists x such thatxis divisible by2 which is true. There exists x such thatxis divisible by4 is true.

$\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$is true.

Thus, $\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$and $\mathbf{\exists }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\vee }\mathbf{\exists }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ are logically equivalent.