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. As a rule, we don't use the passive voice with let.
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We were allowed to do whatever we. In let's get started, the starting point is in view and let's get going, you are on the starting point already. Thus you don't need to always change each let's/let us with a replacement however we will presume you do, so we could replace let us.
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. Moreover, there is a sense. Allow or permit is normally used instead:
