Algebra

# Floor Function

What is $$\left\lfloor\sqrt{46}\right\rfloor?$$

Which of the following statements is/are true?

I. $$\lfloor x+n\rfloor=\lfloor x\rfloor+n$$ for any real $$x$$ and any integer $$n.$$
II. $$\lfloor x\rfloor+\lfloor-x\rfloor=0$$ if and only if $$x$$ is an integer.
III. $$\lfloor x\rfloor+\lfloor y\rfloor\le\lfloor x+y\rfloor\le\lfloor x\rfloor+\lfloor y\rfloor+1$$ for any real $$x$$ and $$y.$$

What is the solution set to $$\left\lfloor \frac{x}{2}\right\rfloor=4?$$

How many integers $$x$$ satisfy the equation $\lfloor\sqrt{x}\rfloor=3?$

What is $$\lfloor \pi \rfloor?$$

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