"there exists" and "for some" are the same, right?

Question

I think "there exists" and "for some" are the same, but still want to make sure.

Ex: $x > ky$ for some k

or there exists a k such that $x > ky$

Answer 1

Yes, they're the same. "There exists an $x$ such that $P(x)$" is the same as "$P(x)$ for some $x$." They would both be written as: $$\exists xP(x)$$