Kenneth Rosen Edition 7 Exercise 1.4 Question 50 (Page No. 56)

Question

Show that $\forall x P(x) \vee \forall x Q(x)$ and $\forall x (P(x) \vee  Q(x))$ are not logically equivalent.

Answer 1

Let $P(x)$ be $x$ passed Physics exam.

and $Q(x)$ be $x$ passed Chemistry exam.

Domain of $x$ be students of a class.

 

$∀x(P(x)∨Q(x))$ will be true if each student either passed the physics exam or the chemistry exam.

 

$∀xP(x)∨∀xQ(x)$ will be true if and only if the entire domain of students either passed the physics exam or the chemistry exam .

Let $\{s1,s2\} $ be the domain of students.

Let $s1$ pass Chemistry exam , $s2$ pass Physics exam.

$∀x(P(x)∨Q(x))$ gives True in this case.

But , $∀xP(x)∨∀xQ(x)$ will give false .

Thus they're logically not equivalent.