a) Show that is logically equivalent to where all quantifiers have the same nonempty domai...

a) Show that is logically equivalent to where all quantifiers have the same nonempty domain.

b) Show that is equivalent to where all quantifiers have the same nonempty domain.

A statement is in prenex normal form (PNF) if and only if it is of the form

where each Qi, i = 1, 2,…, k, is either the existential quantifier or the universal quantifier, and is a predicate involving no quantifiers. For example.  is in prenex normal form, whereas is not {because the quantifiers do not all occur first).

Every statement formed from propositional variables, predicates. T, and F using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact.