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Floor and Ceiling Functions

Floor functions map a real number x to the largest integer less than or equal to x. You can probably guess what the ceiling function does.

Ceiling Function

         

Which of the following statements is/are true?

I. \(\lceil x+n\rceil=\lceil x\rceil+n\) for any real \(x\) and any integer \(n.\)
II. \(\lceil x\rceil+\lceil-x\rceil=1\) if and only if \(x\) is not an integer.
III. \(\lceil x\rceil+\lceil y\rceil\le\lceil x+y\rceil\le\lceil x\rceil+\lceil y\rceil+1\) for any real \(x\) and \(y.\)

What is the solution set to \(\lceil\frac{x}{3}-8\rceil=10?\)

What is \(\lceil\log_{5}59\rceil?\)

How many integers \(x\) satisfy the equation \[\lceil\log_{3}x\rceil=4?\]

What is \(\lceil-\pi\rceil?\)

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