# "Such that" versus "for which"

I was told that such that and for which are not interchangeable, but I cannot determine which one to use. For example,

(a) The collection of numbers x for which x is even.

Or

(b) The collection of all numbers x such that x is even.

I would probably go with (b) instead of (a), but I am not sure why. There are times when for which seems more appropriate, such as the following.

There are subsets of R for which the amount of time the orbit spends in those open sets is not proportional to the sizes of the sets.

When should I use such that or for which and when are they interchangeable?

The reason that (a) sounds odd is that the antecedent of "which" must be "collection," but the closest available noun is "x." What (a) means is

The collection of numbers for which (collection), the members are even.

This refers to a property of the collection. Unfortunately, the proximity of "x" distracts the reader into thinking the property is about "x." Note that your second example excites no discomfort because you're talking about a property of some subsets of R.

The locution

x such that x P

is merely shorthand for "those x with property P," so the phrase "such that" is properly placed.

• I'm still not exactly sure how to tell the difference. Can you provide more examples or link me to more literature on this? Dec 18, 2015 at 22:21
• What is the difference in meaning if I say "There are subsets of R such that the amount of time the orbit spends in those open sets is not proportional to the sizes of the sets." Dec 18, 2015 at 22:22
• I think the literature here is that of naive set theory. You're essentially "translating" symbolic mathematical propositions into natural language. You have to keep your antecedents straight when you translate the bound parameters. Dec 18, 2015 at 23:52
• There's no difference. x="subset of R", P="time not proportional to size"; the antecedent corresponding to the which in for which is x. Dec 18, 2015 at 23:54