Learn English – Which is correct: “gallery for which” or “gallery for whose” or both

As indicated by answers to Can “whose” refer to an inanimate object? on ELU, some people wouldn't be happy with OP's use of it here. I'm not one of them, and given how awkward it would be to avoid the word in OP's construction, I'd just ignore those pedantic prescriptivist grammarians who say it's wrong.

Note specifically Peter Shor's answer on that ELU question, with several examples of Shakespeare using whose to indicate association with inanimate objects.

Here are many thousands of written instances of "equations whose solutions", which is perfectly normal English. By the same token, there's nothing wrong with OP's ...sets of elements whose pairs...

Having (I hope) dismissed criticism of whose, I'd say that most native speakers would use every rather than each in OP's particular sentence. I know I said in another answer that using every in this way is usually a bit stylised/dated. But here it's just "formal", which is appropriate in a "mathematical" context.

I can't exactly explain why "every" is better than "each" here, and I very much doubt it involves any "grammatical rule". But to support my contention that it is "better"...

whose each response (0 hits in Google Books)
whose every response (142 hits)

Also note that the corresponding figures for "whose each/every pair" are 3/116, and for ...solution they're 5/27, which may suggest that mathematicians are less attuned to such subtle distinctions.

There seems to be some disagreement over the distinction between "less common" and "ungrammatical" here, so I'd like to quote from New Scientist - 5 Aug 1989 - Page 55

Rather it elaborates a millionfold democracy whose each unit is a cell.

I've been reading New Scientist every week for decades, and I honestly can't recall ever seeing a grammatical error get through their admirable proofreading procedures. It's also reasonable to assume the authors of Encyclopedia of British Writers, 1800 to Present know their own language...

Golding is an author whose each successive work deals with a different subject in a different time...

EDIT: More recent comments have identified a semantic ambiguity in OP's example that I hadn't originally noticed. I assumed whose referred to "elements" (i.e. - each element within the set contains multiple words, each of which is a false friend to every other word in that element). But apparently whose references "the set", within which each single-word element is a false friend to every other element/word in the entire set.

This is purely an issue of semantics that goes beyond the issue being queried here (whether the usage whose each is "syntactically valid" in OP's example sentence).

Short answer:

the illness FROM which he died - YES

the illness whereof he died - ???

the illness whose he died - NO

my work, the purpose of which is - YES

my work, whose x purpose is - YES

my work, whereof the purpose is - ???

Long answer:

I have never heard "whereof" used in speech. I think it is an extremely formal and possibly archaic term. You do not need to learn it.

I would instead focus on "which" and "whose", which are both important conjunctions.

"which" is very similar to "that". You use "which" when the information presented is optional, and "that" when the information presented is essential and cannot be deleted from the sentence without changing its meaning. Consider these examples from http://www.dailywritingtips.com/that-vs-which/

My car that is blue goes very fast.

If you delete "that is blue" from this sentence, then you do not know which car goes fast. The blue car? The red car? etc.

My car, which is blue, goes very fast.

If you delete "which is blue" from this sentence, the meaning of the sentence does not change. The car color was just extra information that the writer wanted you to know.

"whose" is the possessive form of both "who" (people) and "which" (objects). Here are some examples:

She knew the family whose house we bought.

The chameleon is an animal whose fur changes color.