I think that "Let A be a set, let B be a group, and let C be a number." is the most formal phrasing. Since this is a mathematically formal usage, I think that would be preferred, but I don't think any of the phrases you presented are wrong, and "Let A be a set, B a group, and C a number." is shorter and more succinct. On a separate note, we have a sister site for Math you might consult.
"Let us go/Let us pray" is a special verb construction of the type Let, imperative+us,accusative+bare infinitive. The speaker makes a suggestion and includes himself.
As a rule, we don't use the passive voice with "let". "Allow" or "permit" is normally used instead: We were allowed to do whatever we wanted. We were permitted to drive the vehicle. Accordi...
Let go or be dragged. This is a saying often associated with Zen Buddhism (occasionally Stoicism). As far as I can tell, there's no historical connection, and it might well be originally English. ...
quotes - Origin of the saying "let go or be dragged" - English Language ...
Many people use "let, let's and lets" in conversation What's the difference between them?
meaning - Difference between Let, Let's and Lets? - English Language ...
Let’s is the English cohortative word, meaning “let us” in an exhortation of the group including the speaker to do something. Lets is the third person singular present tense form of the verb let meaning to permit or allow. In the questioner’s examples, the sentence means to say “Product (allows/permits you to) do something awesome”, so the form with lets is correct.