# On the usage of “of” and “for”?

I am writing a report. In the middle of the report, I've written:

Here, in order to reduce complexity we encounter in (33), we use approximation of the X. (The) Approximation for the X is given by:

The question is which preposition is correct, "for" or "of"?

• BTW maybe you need to clean up your articles: "Here, in order to reduce the complexity we encounter in (33), we use the approximation of X. The approximation for X is given by:" Now I need to think why this sounds more natural to me than your original revision . . . – It's Over Jul 31 '15 at 15:28
• @inɒzɘmɒЯ.A.M You are right. it is more sensible since I have already mentioned the approximation. however, sb can say,"an approximation for the X is ..." – Cardinal Jul 31 '15 at 15:52
• We need to know what noun or noun-phrase X stands for. If you don't want to reveal the subject, perhaps you could paraphrase or substitute comparable nouns. – Tᴚoɯɐuo Aug 1 '15 at 10:07

• "...We use an approximation of X."" The approximation we use [for X] is.....,.

square bracketed part is optional, because we already know whst it is an approximation of. But if used, "for" is the appropriate preposition.

Why?

The first sentence says that an approximation OF [the value of] X is created. To approximate something is to make a reasonable estimation of its value.

The second sentence says that that approximation is then used for X (that is, in place of, or standing in for) X.

"of" means "belonging to," or "associated with."

"for" means "on behalf of," or "for the benefit of"

In this case, the approximation is associated with, or belonging to, "the X."

Therefore, "of" is the correct usage.