Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.