Some member of an empty set

Question

Although you may feel my question is somehow strange, please let me ask seriously whether the following 2 statements are true respectively.

Let S be the set of all positive integers less than 0.

Statement P1: For some member x of S, x > 0.

Statement P2: For some member x of S, x = 2 and x is not a prime number.

Thank you in advance.

Symbolically the first statement is $\exists x\in S(x>0)$ or, if you prefer, $\exists x(x\in S\land x>0)$. Does this help? — Git Gud, May 25 '14 at 10:21
The "Related" list at the side of the page here gives a lot of things that aren't quite duplicates but basically answer this question: 1 2 — Ben Millwood, May 25 '14 at 10:37

score 4 · Accepted Answer · answered May 25 '14 at 10:30

A statement of the form "for some member of a set $S$, [...]" is necessarily false if $S$ has no members.

On the other hand, statements of the form "for all members of $S$, [...]" are necessarily true. So you can transform statement P2 into a form which sounds equivalent:

"for all members $x$ of $S$, if $x = 2$ then $x$ is not a prime number"

but is in fact true.

Some member of an empty set

1 Answers1