Algebra
# Floor and Ceiling Functions

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