How can this statement be false? "If $\forall x \in D$, $P(x)$ then $\exists x \in D$ such that $P(x)$."

Question

I'm a college student taking a discrete mathematic course for summer. I took midterm last monday and got back grades and solutions for the exam, but I'm still confused with this specific question.

The question is:

Let $D$ represent a set and $P(x)$ represent a predicate where $x\in D$. Is this a true or false statement? Explain briefly.

"If $\forall x \in D$, $P(x)$ then $\exists x \in D$ such that $P(x)$."

And the answer is: This is a false statement.

How can this statement be false?

( It's late night so I don't wanna bother my professor)

"If every student who bothers the professor late at night fails the course, then there exists a student who has failed." — Blue, Jun 21 '20 at 01:15

score 4 · Answer 1 · answered Jun 21 '20 at 01:11

4

$\forall x\in D,\,P(x)$ can be vacuously true. For example, $\forall x\in\varnothing\,(x>0)$ is vacuously true, since $\varnothing$ has no elements, but $\exists x\in\varnothing (x>0)$ is false, since clearly no such element exists.

answered Jun 21 '20 at 01:11

csch2

4,557

Is this whole thing only false for $D=\emptyset$? – gen-ℤ ready to perish Jun 21 '20 at 01:45
@gen-zreadytoperish If $D$ is nonempty, then $\exists x_0\in D$, and if $\forall x\in D:P(x)$, then $P(x_0)$, so $\exists x\in D:P(x)$. So I believe the only fringe case is when $D=\varnothing$. – csch2 Jun 21 '20 at 04:15
Thank you. So I had to assume that the set was empty? – Lumificier Jun 22 '20 at 16:49
Yes - if $D$ is nonempty then the statement is true, but unfortunately that wasn’t stated as an assumption. – csch2 Jun 22 '20 at 18:22

How can this statement be false? "If $\forall x \in D$, $P(x)$ then $\exists x \in D$ such that $P(x)$."

1 Answers1