For every $x \in A$, $x<5$, there is a number in set $B$ greater than $x$ in mathematical notation

Question

I would like to ask why were my answers wrong. I was supposed to decide which notation has the same meaning as the text (title).

1. $(x<5 \Rightarrow (\exists y \in B)(y>x))$ - This is not correct, because there is not defined that $x$ is from set $A$.

2. $(\forall x\in A)(x<5\land(\exists y\in B)(y>x))$ - This is correct, it is for all x from set $A$, when both $x<5$ and $y>x$ are true, existential quantifier for $y$ from set $B$ is correct either.

3. $(\exists x\in A)(x<5 \lor(\forall y \in B)(y>x))$ - This is not correct, because it is not for every $x$ in set $A$.

4. $(\forall x \in A)(x<5 \Rightarrow(\exists yB)(y>x))$ - I think this is correct, but I said it is not, because $(\exists yB)(y>x)$ is not true just because $x<5$. But I was probably wrong, I am not sure how is it different from 2 in this case.

Thanks for help! :)

Answer 1

In general, "English" is not Precise, hence both (2) & (4) are Possibly Correct !

Interpretation will change the answer.
What is the meaning of "comma" between "$x \le 5$" & "there is a " in the given "English" Statement ?
If that "comma" is "AND" then (2) is correct.
If that "comma" is "then" then (4) is correct.

In either case , (1) & (3) are not correct.

Here is the Example where (2) is correct :
$A=\{1,2,3\} , B=\{0,8\}$
[ all elements in A are $\le$ 5 (AND) those elements have 8 (no worries about 0) in B which is greater ]

Here is the Example where (4) is correct :
$A=\{1,2,3,9\} , B=\{0,8\}$
[ some elements in A are $\le$ 5 (which then implies that) those elements (excluding 9) have 8 in B which is greater ]

Which is Correct ?
We have to know what that "comma" means, to Answer that Question !