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Essendon ... 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.
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Accordi... 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. 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 with We want to see x and y.
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5 "Whether truth can exist without language" and "that truth is an objective reality that exists independently of us" are not opposed claims, although they don't imply one another. A Platonist would tell you that language, like other mental objects, exists in the ideal realm whether people are around to think about it or not. Stay Connected with DON! Keep Up on the Latest Products & Trends! DON. Everything but the Food.
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