Nordquist has a balanced article on jussives that examines different approaches.
I'd say that calling 'Let us pray' an imperative usage is stretching the term somewhat. It is far less hortative than '[Get] on your knees!' In the linked article is:
'[John] Lyons [Semantics, 1977: 747] argues that the imperative can
only be, strictly, second person, and never third person (or first
person). _This may, however, be no more than a terminological issue,
since first and third person 'imperatives' are often simply called
'jussives.'
Types of jussives are also addressed:
Jussives include not only imperatives, as narrowly defined, but also
related non-imperative clauses, including some in subjunctive mood:
Be sensible.
You be quiet.
Everybody listen.
Let's forget it.
Heaven help us.
It is important that he keep this a secret.
...................
In 'Let it be done,' 'it' is referential (with preceding referent) and 'be done' is certainly passive. A modern not-too-near paraphrase is perhaps 'Get on with it, or make sure someone else does!' Far more imperative.
Hmmf, well the question used to ask if someone could prove if this phrase was grammatical but this response inspired a edit removing that request for a proof. So what follows now seems a bit silly. I invite the first editor so inclined to delete this down to where it says "regarding edit".
Yes, I will be proving that someone can prove "Yes, I will be." is grammatical. I will do this by making up a grammar where the string "Yes, I will be." is the only grammatically correct string in the whole grammar. This makes proving it to be grammatical a tautology. It must be correct since it is how the grammar was defined. It's correct by definition.
Now, if you find that unsatisfying I invite you to cite a grammar definition that is formal enough to withstand the very idea of proofs. Proofs are rigorous but only as good as the context from which they emerge.
Otherwise we're just having a subjective argument. One that I'll win, only because my hat looks cooler. :)
As a side note, the only objection I know to this common usage is that it ends with a preposition. And we all know how Churchill felt about that.
Regarding edit:
- Yes, I will be
Why is answer no.3 grammatical?
It's not. Ya left off da period ya silly. No wait, that's punctuation. Yeah it's fine, if it means what it says. In fact I can imagine this exact phrasing being the point of the answer:
CandiedOrange took a piece of cake over to Mary while she was sitting with friends who were watching her unwrap gifts. CandiedOrange asked, "Will you be having cake?" Mary narrowed her eyes at this untimely interruption and responded, "Yes, I will be".
What evidence is there to support it?
Speaking of my hat, when Luke says "I'm not afraid" Yoda says, what to me sounded like the only non-mangled English he ever spoke, "You will be".
So don't go disrespecting my childhood hero. : )
- Yes, I will do.
Is answer number 4 (above) ungrammatical?
This is what I mean by 'if it says what it means': "Yes, I will do (nicely)." is perfectly grammatical. It also has nothing to do with the question:
Will you be having cake?
At least, not without some very convoluted backstory where you ARE the cake you're eating. I've read weirder stories, but that way lies madness.
What I'm pointing out here is that the question imposes a subset of possible meanings. There are ways to parse these words that result in them being grammatical but in the context of the question those grammatical meanings are nonsensical. So the answer is wrong. Just for a different reason.
Which means it's not the best answer to use in this question. You want something objectively ungrammatical. This is about English, not logic or philosophy.
So I vote for a new answer:
Yes, I will a.
If you know how to parse that into any meaning at all, you're more clever than I.
Best Answer
Under exactly the conditions that you have in the question.
Whose is used as a relative pronoun, to introduce a clause that describes something belonging to the noun phrase it follows.
Now, there are some people do object to the adjective senses of whose being used of inanimate antecedents, based on the mistaken belief that it comes solely from who, when it comes from whos which comes from hwæs which was the genitive of both hwa (who) and hwæt(what). The objection flies in the face of much eloquent usage, and is also rarer now.
Each is a determiner, it refers to all of the examples of the thing that will be named, so that what is stated of them singularly applies to every one of them.
We have "each pair is". Generally, those things covered by each are treated as singular, unless each follows a plural subject, in which case they are treated as plural, with a further exception allowing (some would say not, while some would say insisting, it's here we enter into a matter of debate) if the plural subject is the pronoun we.
Here we do not have each following a plural subject, so the singular should be used, and so it is. Again, we're fine.
Putting them together we have a noun phrase "The set of elements" followed by a relative pronoun, followed by a determiner which insists upon singular use of what follows, followed by a singular use.
It is perfectly grammatical. Looking for objections to different uses of the words involved, we find that not only are there none, there aren't even questionable objections to argue against.
Now, it is relatively rare. It's common here to use where instead of whose, which ironically is a use that does frequently find objectors saying where can only refer to place, literally or figurative.
It's common to use every or all. However each conveys a sense of precision; it's merely a side-effect of every being treated plural and each being treated singularly leading to a sense that we are focusing on each item rather than making a more sweeping statement, and there's no real lack of precision with every, but that impression is worth making in technical cases like mathematics.
And for that reason, it is relatively common in such contexts.
About the only possible objection I can see to this as a whole, is that the form "[Noun]1 of [Noun]2s whose each [Noun]3 is..." could be ambiguous as to whether it is [Noun]1 or its [Noun]2s that possesses the [Noun]3s that are being described.
However:
So that possible objection clearly doesn't apply here. It's also not a question of grammaticality.
In all, the form is not just grammatical, and reasonably common in the domain it is used in, but a good choice.
[Taking a look at the question on ELL, it seems that the problem was that it wasn't ambiguous enough, as what they meant was "set whose each pair of elements", which is neither of the readings I suggest are possible with the form, but that's a separate issue].