A number that is even and prime.

Question

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!

Answer 1

"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.

Answer 2

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 $\operatorname{Prim}(x)$".