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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.

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1 Answer 1

up vote 2 down vote accepted

Your arguments are perfect.

As you stated, $\forall x:Q(x)\to P(x,y)$ becomes automatically true if $Q(x)$ is never true, and $\exists x:Q(x)\land P(x,y)$ becomes automatically false if $Q(x)$ is never true.

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