Find the possible meanings of the formula:(∀x∈A)((x∈B)⇒(∃k∈ℕ)(∃n∈ℕ)(x=k⋅n)) I'm doing this task where I should find the possible meanings of the formula:(∀x∈A)((x∈B)⇒(∃k∈ℕ)(∃n∈ℕ)(x=k⋅n)). From these options:

*

*All elements that are simultaneously in A and in B, we can write in the form of a product of natural numbers.


*Each element of the set A is in the set B and can be written as the product of two natural numbers.


*For each element of the set A two natural numbers can be found n and k, so that this element is their product.


*For each element of the set A, which is also in the set B, we can find two natural numbers k and n, so that this element is their product.
I thought that it is the third one but I was probably wrong. There could be zero or many right answers. I've been doing this for 3h and will be glad for any help, advice, anything. Thank you in advance.
 A: Option 4 is correct. Option 1 should also be correct, but it has been worded a little strangely. If you ask me they are both correct, but if only one of the answers is supposed to be correct, then pick option 4.
Let me elaborate on how I reached this conclusion. We have the predicate
$\forall x\in A(x\in B\implies \exists n,k\in \mathbb{N}:x=k\cdot n)$
The statement gets "translated" like this: "For every x in A, if x is (also) in B, then there exist two natural numbers k and n, such that $x=k\cdot n$". As you can see the wording is nearly identical to that of option 4, the only difference being that in option 4, rather than giving a "name" (x) to the element of set A, it is just called "this element" instead, which admittedly can lead to confusion.
However, the predicate can also be rewritten like this
$\forall x(x\in A \implies(x\in B\implies \exists n,k\in \mathbb{N}:x=k\cdot n))$
Which is equivalent to
$\forall x((x\in A \land x\in B)\implies \exists n,k\in \mathbb{N}:x=k\cdot n)$
This is read as "for every x (in option 1, for every element) that is (simultaneously) in A and in B, there exist two natural numbers whose product is equal to x." Presumably, the second part of the sentence should still preserve its meaning as you word it as "x can be written as a product of two natural numbers" instead.
On second look, option 1 reads simply "as a product of natural numbers", not "as a product of TWO natural numbers" specifically. Because of this, option 1 should also be incorrect.
Option 2 is incorrect because nowhere in the predicate is it stated that every x in A is also in B. The predicate only gives information on those elements of A that HAPPEN to also be in B.
Option 3 fails to mention that for the implied statement to be true, x must ALSO be in B, not only in A.
I hope I was able to help!
