Are these two logical formulas logically equivalent:
∀x ∈ U(P(x) ∨Q(x)), and ¬(∃x ∈ U(¬P(x)) ∧ ∃x ∈ U(¬Q(x))), where we let U be a domain.
For the 2nd statement, I simplified the expression
¬(∃x ∈ U(¬P(x)) ∧ ∃x ∈ U(¬Q(x)))
= ¬∃x ∈ U(¬P(x)) v ¬∃x ∈ U(¬Q(x) )
= ∀x ∈ U(P(x)) v ∀x ∈ U(Q(x) )
= ∀x ∈ U(P(x) v Q(x) )
Therefore I assumed the 2nd statement would be equivalent to the 1st statement, but the answers to this question says it is not. Where did I make a mistake in my simplification procedure, or is there another reason why we cant consider these 2 statements equivalent? Thank you for your support in advance.
Note: Your notation is a bit weird I'm not sure where you get it on but I'll adapt to it
The problems lies here: this is not true. A simple example is letting x be a natural number and P(x) be "x is odd" and Q(x) be "x is even"
The first predicate is "Either all natural number are odd, or all natural numbers are even"
The second predicate is "Every natural number is either odd or even"
Clearly that the first one is wrong and the second one is right in this case
The correct answer is
$$\forall x \in U(P(x)) \lor \forall x \in U(Q(x))\\ =\forall x,y\in U((P(x) \Leftrightarrow P(y))\lor (Q(x) \Leftrightarrow Q(y)))$$