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7

The formal statement ∃ x ∈ A : P(x) using the existential quantifier reads as Es existiert ein x ∈ A, so dass P(x) gilt or Es gibt ein x ∈ A, so dass P(x) gilt. It is unnecessary to say mindestens ein or something like that. You can also say P(x) für ein x ∈ A. or Es gilt P(x) für ein x ∈ A. Saying für irgendein would be highly unusual. If you really ...


4

Your unease is justified, “irgendein” would be unclear, as it could be read as “any” and hence universal quantification. At least it would be so unusual that you should not use it. That aside, you need to decline, but luckily all letters are neuter, so “für irgendein x” is grammatically correct. Unfortunately, it has been a while since I have written any ...


5

From my days with formal math at school and university I remember wordings like Es existiert mindestens ein x ∈ A, für das gilt: (...)


2

You could use "irgendein", but it would come across as a bit colloquial. The shortest wording known to me would be P(x) für ein x aus A I'd say that it would be used among mathematicians discussing a problem, but it is only valid in that discussion due to context. To explicitely state the arbitrariness, I'd say: P(x) für ein beliebiges x aus A ...


2

Yes, it is correct, and it is in my opinion the clearest way to state the fact if you want to avoid the lengthier "Es existiert ein x, für das P(x) gilt". Other possibilities are "für ein x ∈ A" or "für ein geeignetes x ∈ A" (as suggested by Carsten S). Do not use "für x ∈ A" or "für ein beliebiges x ∈ A"; ...


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