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How to write "There is v in set W and P, such that v is in P"?

I thought of the following possibilities:

∃ v ∈ W : ( ∃ P : v ∈ P)

or

∃ v ∈ W, P : v ∈ P

Is either of these correct? If not, how should I write it?

Andrés E. Caicedo
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Eduardo Bezerra
  • 131
  • Your title does not match what you asked. Which is the intended question? – MPW Dec 01 '15 at 13:41
  • You're right. Sorry about that. I just couldn't write the ∄ symbol, so I switched to ∃. – Eduardo Bezerra Dec 01 '15 at 13:46
  • Without some constraints on $v$, $W$ and $P$ both sentences are obviously true: for each $v$, despite belonging to $W$ or not, there exist such $P$, that $v\in P$ — namely a singleton $P={v}$, and there exist such $Q$ that $v\notin Q$ — for example the empty set $Q={}$. – CiaPan Dec 01 '15 at 13:46
  • @CiaPan, You're right, but that's irrelevant to the question... I just want to know how to represent the idea with mathematical symbols... – Eduardo Bezerra Dec 01 '15 at 13:48
  • @EduardoBezerra That gets an answer. This got a comment. – CiaPan Dec 01 '15 at 13:51

2 Answers2

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This should suffice: $$\exists v \in W : v \in P$$ Another, equivalent expression is: $$\exists v \in W \cap P : v \in P$$

Björn Friedrich
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You can write 'there is no such P' in several ways. Directly $$\not\exists P : v\in P$$ or 'it's not true there exist such P': $$\neg\left(\exists P : v\in P\right)$$ or even by transferring the negation into the predicate, as 'for each P, v can't belong to P': $$\forall P : v\notin P$$

CiaPan
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