In English one can write the existential statement ∃ x ∈ A : P(x) as follows:
P(x) for some x ∈ A.
Is there a similar construction in German? My best guess is:
P(x) für irgendein x ∈ A.
- Is this use of irgendein correct? Does it correctly reflect the meaning of existential quantifier?
- Does irgendein need to be declined in this case and, if so, how? I don't know what grammatical gender x should take. Is it das Element x, die Variable x or der Buchstabe x ?
- How would it be for more than one variable, e.g. ∃ x,y ∈ A : P(x) ?
EDIT
To clarify: It's important to distinguish here between a universal statement and an existential statement. I'll be more specific about why I would like a translation of the "for some" construction by providing a concrete example:
Let B = {(n,2n) : n ∈ N} and let b ∈ N. Then b = 2m for some m ∈ N.
Here m is not an arbitrary element, since it depends on b. I want to avoid constructions such as "Es existiert m ..." because I want to introduce b before m in a concise way.