"And" or "Implication" in this case?

Question

I want to say that:

There exists a subset $X$ of $\mathbb{N}$ for which $|X|$ $= 5$.

Can someone explain to me why one would choice either of these three forms over the other ones?

• ($\exists$$X$)(($X \subseteq$ $\mathbb{N}$)($|X| = 5$))

"There exists a $X$ such that $X$ is a subset of the natural numbers such that its cardinality is equal to 5."

• ($\exists$$X$)(($X \subseteq$ $\mathbb{N}$) $\implies$ ($|X| = 5$))

"There exists a $X$ such that $X$ is a subset of the natural numbers implies that its cardinality is equal to 5."

• ($\exists$$X$)(($X \subseteq$ $\mathbb{N}$) $\land$ ($|X| = 5$))

"There exists a $X$ such that $X$ is a subset of the natural numbers and that its cardinality is equal to 5."

Answer 1

In sentences like "there exists something such that this and that", "something" is usually meant as a member of some set, in math jargon: this makes you write $$(\exists x \in X)\mathcal{P}(x)$$ But this is nothing more that a shorthand for the formula $$(\exists x)(x\in X\land\mathcal{P}(x))$$

The same happens for the universal quantifier: sentences like "for all things, this and that", i.e.: $(\forall x\in X)\mathcal{Q}(x)$, are only abbreviations that stand for $(\forall x)(x\in X \implies \mathcal{Q}(x))$.

Answer 2

It's the third one. The first one fails because it has no logical connective in it. You are writing $P(X)Q(X)$ and reading it as if it said $P(X)\wedge Q(X),$ but we wouldn't need the $\wedge$ symbol at all if this were legitimate.

As to the second one, the statement you have written says that if $X$ is a subset of $\mathbb{N},$ then $X$ has cardinality $5,$ that is to say, all subsets of the natural numbers have $5$ elements.