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.

Understanding the Context

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.

Key Insights

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. Note: If you don't see the security questions option, it's possible that your organization doesn't allow you to use this option for your password reset method. In this case, you'll need to choose another method or contact your organization's help desk for more help. If you are unable to use the account recovery form because you don't recognize the additional account security information, use our Sign-In Helper. To protect your account and its contents, our support agents and advocates are not allowed to send password reset links or access and change account details.

Final Thoughts

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