The structure of Noun Phrases
Simple noun phrases usually have two sections, a Determinative and a Head. The Determinative function is typically carried out by special words called DETERMINERS. The Head function is usually carried out by a phrase headed by a noun. This type of phrase we see inside a noun phrase is called a NOMINAL.
In the phrase a big dog, we see the following structure:
Now, because the nominal is a proper phrase in its own right, it has its own internal structure. Like all phrases, it has its own Head. It also very often has a Modifier. When this Modifier comes before the head, we call it an Attributive Modifier. Attributive Modifiers are very often adjectives. In the phrase big dog, the Modifier is the adjective, big, and the Head is the noun, dog.
Now, notice that I said that Determinatives are "typically" determiners, and that Modifiers are "often" adjectives. This is because other types of words and phrases can carry out these jobs. For example in the noun phrase:
... the Determinative is the genitive noun phrase John's, not a determiner. Similarly, the Modifier in the nominal is the proper noun, Madonna, not an adjective.
Notice also that both determiners and adjectives can carry out other functions apart from Determinative, or Attributive Modifier:
- That was the fastest I ran.
- Indignant, she waltzed out of the room.
Each versus Every
One of the other answers here brings up a comparison with each and every. At first blush, this seems quite reasonable. Firstly, both each and every are determiners. They also have some similarities in terms of meaning. For example, we can say that they sometimes give us a meaning somewhat similar to all:
- All the men have finished their food.
- Every man has finished their food.
- Each man has finished their food.
The sentences above basically give us the same kind of information. There are no men who haven't finished their food. Notice from the sentence above that each and every have some similar grammatical properties too. For instance, they both take singular nouns: in the examples above we see "all the men", but "every man" and "each man". This is because every and each have DISTRIBUTIVE meanings. We consider that the verb phrase finished their food applies to each member of that group of men considered individually.
Another difference you might have noticed is that in the first example, all occurrs with the determiner the. The word all differs from prototypical determiners because it can occur with a different function in a special slot before the Determinative. Writers like Quirk et al (1985), call words like all "pre-determiners".
Determiners like each and every, the, a, this, no and also genitive noun phrases such as his, ones and Ben's have an important property: we can only have one of these types of words in Determinative function. The following phrases where we find two such words in Determinative function are therefore badly formed:
- *The my dog is wagging its tail.
- *A my friend is coming.
- *The every man finished their meal.
- *Each the book was shelved properly.
- *No Ben's friend came to the party.
So we've seen that each and every have some similar properties: they're both determiners; semantically they have distributive meanings similar to all; they cannot occur in determinative function together with other central determiners or with genitive noun phrases.
However, at this point all similarity ends. Each and every have other semantic and syntactic properties which make them very different. We shall see that just because we can use every in a particular way, it does not mean we can therefore use each in that same way - or vice verse.
First of all, we can use each to stand in for an each + [noun] combination. We can't do this with every:
- Give me one of each.
- *Give me one of every. (ungrammatical)
Because of this we can use each on its own as an Adjunct after the Subject in sentences such as:
- The baboons each gave the elephant a donut.
- *The baboons every gave the elephant a donut.
We can also therefore use each and not every on their own in partitive expressions:
- Each of them ...
- *Every of them ...
A second difference that's important in terms of syntax is that we can freely modify the word every, but not the word each:
- *Almost each dog ate a bone.
- Almost every dog ate a bone.
In the first example we see each modified by the word almost with bad results. The example with every is fine.
There are many, many more differences between each and every, but, for the moment, I'll mention just one more: we can use every without a distributive meaning, mostly with abstract nouns. When we use it like this, it has a "multal" meaning. In other words it means something like "a lot of":
- We gave him encouragement.
- We gave him every encouragement.
- *We gave him each encouragement.
We can see the non-count use of encouragement in the first example above. In the second example we see the same non-count use of the word encouragement. This example does not mean that there was a group of encouragements and we gave him all of them. It means we gave him a lot of encouragement or a variety of encouragement. The last example cannot mean the same thing. Encouragement here cannot be that non-count sense of encouragement. The determiner each cannot have a multal instead of distributive meaning.
The Original Poster's question
*The set of elements whose each pair is ... (ungrammatical)
The sentence above is ungrammatical. The reason is, simply, that whose is a genitive noun phrase, just like his or Ben's. We have already shown further above that each and every cannot be used together with genitive pronouns in Determinative function. I fact whose, like other genitive noun phrases, cannot be used with central determiners as Determinatives at all:
- *A man whose the dogs bit him went to the hospital yesterday.
- *The dictionary whose a page you tore out is missing.
- *The elephants whose their donuts you ate has escaped.
- *That man whose every toe was amputated went to the police.
- *The wife whose each husband died in mysterious circumstances absconded.
The sentences above are all badly formed. They each try to use whose with another genitive noun phrase or determiner as a Determinative.
Some of the other comments here have noted that we can use whose with every in phrases like:
- whose every move (grammatical)
We can also use every after other genitives such as his or her:
These posters argue that if whose every move is grammatical, then whose each move must also be grammatical. This does not follow though, because as we have shown, each and every do not have the same grammar. However, in any case this really isn't the key issue here anyway. The Original Poster's example is clearly asking about using whose each as a Determinative. We have shown that this is not possible.
The phrase whose every move does NOT use every as a Determinative, it uses it as a Modifier in the nominal every move. The word each cannot appear alone in this function. The structure for the phrase whose every move is:
As the Cambridge Grammar of the English Language (Huddleston & Pullum, 2002) states in its section on every versus each:
(g). Every, but not each, can occur as a modifier following a genitive [...].
 a. They scrutinised [her every move]. b. * They scrutinised [her each move].
The [ * ] next to example (b) there indicates that it is ungrammatical. Example (a) is structured [her] [every move]. It is similar in meaning to "[every move] of hers".
This clearly shows that we cannot use each as a modifier after genitives such as her or whose. (We can however use the idiomatic phrase each and every, which patterns the same as every here).
I'm assuming that readers know that we can use relative whose to refer to both people and object. I won't address this here, Fumblefingers has links to relevant posts on that subject in his answer, if you'd like to read about it.
I use the term Determiner in this piece, like the majority of modern grammars, to refer to the class of words. I use Determinative to refer to the function. Huddleston and Pullum reverse the established terminology in CGEL.
John Hanna and Fumblefingers state in comments or posts that for them whose each is grammatical. It might be the case that whose each is grammatical in the OP's intended usage for speakers of some varieties of English. I'm most familiar with so-called Standard GB, and it might be that whose each is grammatical for some other groups of speakers.
Man_From_India has suggested that phrases such as supported his each step and similar which might be grammatical. I don't think this is so. It is possible to find a few examples from Googlebooks (40 at first glance). However, these turn out mostly to be running across sentences ["his. Each step ..."], or else they are self-published unedited texts full of typos.Here's the Ngrams graph for his each step and his every step. As you'll see there are no results at all for his each step: