Question

Show that \(\forall xP(x) \cup \forall xQ(x)\,\,and\,\,\forall x\forall y\,(P(x) \cup Q(y)),\)where all quantifiers have the same nonempty domain, are logically equivalent. (The new variable y is used to combine the quantifications correctly.)

Short answer

The two statement where all quantifiers have same nonempty domain are logically equivalent can be shown.

Step-by-step solution

Stating true statement

Suppose that the true statement is \(\forall xP(x) \cup \forall xQ(x).\)When P(x) and Q(x) is true for all x. when P(x) is true, then \(P(x) \cup Q(y)\)is true for all x and y. Thus, true is \(\forall x\forall y(P(x) \cup Q(y)).\)

Proving the statement

Suppose \(\forall x\forall y(P(x) \cup Q(y))\) is true. When P(x) is true then \(\forall x(P(x) \cup Q(y))\)also true. When P(x) is false then x,Q(y) is true for all y. When Q(y) is true then \(\forall xP(x) \cup \forall xQ(x)\)also true. The statements are logically equivalent.