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I am a little bit confused about the meaning of quantifier ∃x

Some definitions make it sound like There is exactly one x and some make it sound like There is at least one x

I have this expression:

∃x (P(x)∧ Q(x))

Does it mean find exactly one x for P(x) and Q(x) or does it mean find one or more x for P(x) and Q(x)?

Boris Grunwald
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    It means that at least one such $x$ exists that makes $P(x)$ and $Q(x)$ true. – SescoMath Sep 20 '18 at 11:49
  • "at least one"... – Mauro ALLEGRANZA Sep 20 '18 at 11:49
  • The latex notation is $\exists , ! , x$ instead of just $\exists , x$. – Dietrich Burde Sep 20 '18 at 11:52
  • @DietrichBurde. You should say that the exclamation mark is used to denote "exactly one", i.e. $\exists!$ denotes "there is exactly one". – md2perpe Sep 20 '18 at 12:00
  • @DietrichBurde. But the meaning of $\exists x$ is not "there is exactly one", but "there is at least one". – md2perpe Sep 20 '18 at 12:06
  • However, too prove "$\exists x : P(x)$" it is enough to find one $x$ such that $P(x)$. Too prove $\exists! x : P(x)$" one should find one $x$ such that $P(x)$ and then show that if also $P(y)$ then $x=y$. – md2perpe Sep 20 '18 at 12:08
  • @DietrichBurde. The question was about the meaning of $\exists x$. – md2perpe Sep 20 '18 at 12:10
  • @DietrichBurde. That doesn't change much. Yes, you only need to find one $x$ to prove $\exists x$. But that doesn't mean that $\exists !$ should be used instead, since to prove the $\exists !$ you also need to prove uniqueness. – md2perpe Sep 20 '18 at 12:15
  • No, I just mean that for a unique $x$ (he was asking "Does it mean find exactly one x for P(x) and Q(x)") you need to use $\exists , ! , x$, and not $\exists , x$. – Dietrich Burde Sep 20 '18 at 12:16
  • @DietrichBurde. What you meant wasn't obvious. And what does "find exactly one" mean? Does it mean that you just need to find one but there might exist more, or does it mean that you need to find one and should prove that there are no more? – md2perpe Sep 20 '18 at 12:33

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Exactly one is often denoted as

$$\exists!$$