Let $A=\{6,10,30\},B=\{3,5\}$ and $P(x,y)=$ $x$ is divisible by $y$
State whether it is true or false for the below statements.
1.For any odd integer $x$ in $A$, for any $y$ in $B$, $P(x,y)$
This statement is true, because it is a vacuous truth statement. Since it is of the form $∀ \ odd\ x \ \exists \ y$, and there are no odd $x$ in $A$.
2.For some y in B, for any odd integer x in A,P(x,y) y odd
This statement is is true as it is also a vacuous truth statement. It is of the form, $\exists y\forall odd\ x$. The existence of y 3, 5 and there are no odd x in A.
3.For any odd integer x in A, for some even integer y in B,P(x,y)
I am not too sure about this one... This statement is true because there are no odd integers in A.
4.For some even integer y in B, for an integer x in A,P(x,y)
This statement is false, because there are no even integers in B, and we can't use for some.
Check my answer thanks! Also, is there a better way to state the justification more concisely and precisely? Generally, I feel I am plain confused about All and Some statements for empty sets.