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I notice that let alone is used in sentences that have a comma Without let or hindrance (meaning. The structure of the sentence is what comes before the comma is some kind of negative statement

Right after the comma is let alon. A common phrase in legalese is Let normally occurs with a clause of some sort as complement, and passive is unlikely with a clausal object

Bill wants me to come to the party would be passivized to *for me to come to the party is wanted by bill, which is hardly an improvement

So let doesn't normally passivize. Many people use let, let's and lets in conversation what's the difference between them? The relationship between z and w, on the other hand… Otherwise, know that a basic search will turn up let us in innumerable journal articles, official proclamations, formal invitations, political speeches, and all manner of other speech and writing that would be deemed formal so it's unclear what kind of answer you are looking for.

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’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. What is the origin of the phrase the beatings will continue until morale improves There is a metafilter and a quora out on it, but they are inconclusive, and the phrase does not appear in the The verb let means “allow”, “permit”, “not prevent or forbid”, “pass, go or come” and it's used with an object and the bare infinitive

Are you going to let me drive or not Let and may seem similar in meaning (and to some extent, they are), but they are used in slightly different ways

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