Solution to Show that ∀xP(x)∧∃xQ(x) is logically equivalent to ∀x∃y(P(x)∧Q(y)), where all quantifiers have the same nonempty … - Sikademy
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Archangel Macsika

Show that ∀xP(x)∧∃xQ(x) is logically equivalent to ∀x∃y(P(x)∧Q(y)), where all quantifiers have the same nonempty domain.

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Here's the Solution to this Question

We will start from the first given statement,


∀xP(x)∧∃xQ(x)


It doesn't matter if the variables are called x or called y, thus let us recall the variable in the second expression of the conjunction:


\equiv∀xP(x)∧∃yQ(y)

\equiv(∀xP(x))∧(∃yQ(y))

\equiv∀x(P(x)∧(∃yQ(y)))


Using the commutative law,

\equiv∀x((∃yQ(y))∧P(x))

\equiv∀x∃y(Q(y)∧P(x))


Again using the commutative law,

\equiv∀x∃y(P(x)∧Q(y))


Hence we can say that the statement ∀xP(x)∧∃xQ(x) is logically equivalent to ∀x∃y(P(x)∧Q(y)) where all quantifiers have the same nonempty domain.


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