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Prove or disprove each of these statements about the floor and ceiling functions.

  1. $\left \lceil \left \lfloor x \right \rfloor \right \rceil = \left \lfloor x \right \rfloor$ for all real number $x.$
  2. $\left \lfloor 2x \right \rfloor = 2\left \lfloor x \right \rfloor$ whenever $x$ is a real number.
  3. $\left \lceil x \right \rceil + \left \lceil y \right \rceil - \left \lceil x+y \right \rceil =0$ or $1$ whenever $x$ and $y$ are real numbers.
  4. $\left \lceil xy \right \rceil = \left \lceil x \right \rceil \left \lceil y \right \rceil$ for all real numbers $x$ and $y.$
  5. $\left \lceil x/2 \right \rceil = \left \lfloor x+1 / 2 \right \rfloor$ for all real numbers $x.$
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