Is $\forall x(x\in\varnothing)$ true or false? I know that
$$\forall x(x\in\varnothing)$$
is equivelent with

"There is no element in Universe such that $x$ is not in $\varnothing$."

but, is $\forall x(x\in\varnothing)$ true or false?
Is it an example of vacuously true?

How about the Univers is the empty set?


How about the Univers is not the empty set?

 A: $x$ refers to an element of a Set. You need to first define the set to be able to make propositions with it's elements. Your expression is not well defined, meaning, there is not a unique correct answer.
Here is an analogy :

If i asked you how much a car costs. Your answer is probably : I don't know, depends on the car. Also depends on where you buy it. Depends on what you mean by "cost", do you include insurrance ?

Therefore, "How much does a car cost ?" is usually not a well defined expression. Same holds true for your question.
So let's give your question some context. First, what will we call a set ?
You can (naively) define a set by :

*

*Listing all of it's elements.

*A property.

Saying : "$\forall x \,: \, (\text{something is true)}$" without specifying a Set, from which to draw $x$ from, wether it be from prior context or explicitly, is in many ways the same as asking that car question.
So let's try to give it context :
Turns out that there are only two possible contexts to interpret your proposition. Or in other words, two ways to make it well-defined. A set is either empty, or it is not. Let's call our Set $X$ :

*

*If $X$ is empty, your proposition becomes : $\forall x \in X : \:(x \in X)$, which is true, by definition of $x$.

*If $X$ is not empty : Notice that $\emptyset \subset X$.Also, for any $Y\subset X$, The following proposition : $\forall x \in X : \:(x \in Y)$ is always false, unless $X=Y$. Since by assumption, $X$ is not empty, we have $X \neq Y$ and thus your proposition is false.

Now, regarding vacuously true. You first need a well defined expression, in order for something to be "vacuously true". And indeed, propositions on the empty set, are "vacuously true". But like i said, they need to be well defined first.
A: It is false.
Definition: By axiom, we have $\emptyset$ such that $\neg \exists x(x\in \emptyset)$.
Theorem: $\neg \forall x (x\in \emptyset)$
Proof:
Suppose to the contrary that $\forall x (x\in \emptyset)$.
Substituting $x=\emptyset$, we would then have $\emptyset \in \emptyset$ and a contradiction:  $\neg \exists x(x\in \emptyset) \land \exists x(x\in \emptyset)$
Therefore,  $\neg \forall x (x\in \emptyset)$.
A: 
$$\forall x(x\in\emptyset)\tag1$$

Translation: Every member of the universe belongs to the empty set.

*

*If the universe is not actually empty, then any member of it belongs
to a nonempty set, so sentence $(1)$ is, by counterexample, false.


*On the other hand, if the universe is $\emptyset,$ then, given any
universal sentence $\forall x\,\psi(x),$ no counterexample can
possibly be found to falsify it, so $\forall x\,\psi(x)$ is true
and, in particular, sentence $(1)$ is true.
Hence, sentence $(1)$ is true precisely when the universe of discourse is $\emptyset.$ Think of the sentence as characterising an empty universe of discourse.
And, since, in an empty universe, every universal sentence is vacuously true, sentence $(1)$ is vacuously true precisely the universe of discourse is $\emptyset.$
(So, since sentence $(1)$ is false when the discourse domain is, say, $\mathbb R,$ it is not a logical validity.)
A: No. Vacuous truth arises only when the unfulfillable statement occurs as the restriction in a restricted universal quantification, i.e. as $\psi$ in in $\forall v (\psi \to \phi)$, because then there are no objects that could satisfy $\psi$ but not $\phi$ and thereby falsify the implication.
When it is the main predication itself of the universal quantification that fails to be satisfied, i.e. $\phi$ in $\forall v (\phi)$, then the universal claim is false. It is not the case that all elements of the universe are an element of the empty set, in fact none of them are, so the statement is simply false.
The universe of a first-order-model is, by definition, always non-empty.
A: The empty set is the set such that $x\notin\emptyset$ for any $x\in X$ where $X$ is some set or another. The confusion arises when you say "for all $x$ in $X$", because if $X$ has no elements, in other words if $X=\emptyset$ then it is true that "for all $x$ in $X$, $x$ is in the empty set" since there is no $x$ in $X$.
It is like me saying that "All of my Aston Martins are covered in gold leaf" a true statement, but meaningless since I don't own an Aston Martin. My friend Al does own an Aston Martin, but it's not covered in gold leaf, so he can't truthfully claim that all of his Aston Martins are covered in gold leaf, whereas I can.
