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So I have the following sentence : There exists a unique prime number that is even.

I have the following predicates :

Prim(x) : x is a prime number. Even(x) : x is an even number.

My answer : Prim(Even(x))

I don't know, but I think that this would be better : Ǝ Prim(Even(x))

Would this make sense ? Even though we know that there is only one such number that is prime and even.

Thank yoU!

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    The expression "Prim(Even(x))" is not well-formed: it's basically saying "'$x$ is even' is prime." – Noah Schweber Sep 08 '15 at 01:28
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    You need to say that there exists a number that is both even and prime (existence), and any number that is both even and prime is equal to that number (uniqueness). Can you see how to formalize that? – mjqxxxx Sep 08 '15 at 01:30
  • Yeah, so I would get this : Prim(x) ∧ Even(x) But How do I know that there aren't more than one ? How do I make sure that this represents only one particular number ? –  Sep 08 '15 at 01:32
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    Hint: for any $y$ with the same property follows $y=\dotsc$ ? – user251257 Sep 08 '15 at 01:44
  • Sorry user, I didn't see this type of stuff. I only know the really basic stuff about it... I'm not sure how to make Prim(x) ∧ Even(x) unique.... –  Sep 08 '15 at 02:16
  • @Astroman unique (or more precisely at most one) means, if $x$ and $y$ are even and prime, it follows $x=y$ – user251257 Sep 08 '15 at 02:45
  • Prim(x) ∧ Even(x) --> Equal(x,2) DOes this make sense ? To be honest, I have no idea.... –  Sep 08 '15 at 03:35

2 Answers2

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"There is exactly one even prime number" can be expressed formally as $$ \exists x: [(Even(x) \land Prim(x)) \land \forall y : [Even(y) \land Prim(y) \implies x=y]]. $$ The first part says there is at least one even prime number. The second part says there is at most one even prime number.

user251257
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For the statement "There exists a unique Prime number that is Even." it's rather more simple.

$\exists x: [\operatorname{Even}(x)\land\operatorname{Prim}(x)]$

My explanation for this is that the existential quantifier $\exists x$ means:-

  1. There exists an 'x'
  2. For some 'x'
  3. For at least one 'x'

Here it is used to reference that "there exists a unique 'x' satisfying both $\operatorname{Even}(x)$ and $\operatorname{Prim}(x)$".

abhinit21
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