Proposition: Properties of Floors and Ceilings 

The floor and ceiling functions have the following properties:

1. $\lfloor x \rfloor\le x $ and $\lceil x \rceil \ge x$ for all real numbers $x\in\mathbb R.$
2. $\lfloor x \rfloor = x $ and $\lceil x \rceil = x$ if and only if $x\in\mathbb Z$ is an integer.
3. If $x\not\in\mathbb Z$, then $\lceil x \rceil-\lfloor x \rfloor=1,$ otherwise $\lceil x \rceil-\lfloor x \rfloor=0.$
4. For every integer $n\in\mathbb Z$, $\lfloor x+n \rfloor = \lfloor x \rfloor+n.$
5. In general, $\lfloor nx\rfloor\neq n\lfloor x\rfloor.$
6. Redundant floor and ceilings brackets:
   $(a)$ $x < n\Leftrightarrow \lfloor x\rfloor < n,$
   $(b)$ $n < x\Leftrightarrow n < \lceil x\rceil,$
   $(c)$ $x \le n\Leftrightarrow \lceil x\rceil\le n,$
   $(d)$ $n \le x\Leftrightarrow n\le \lfloor x\rfloor.$
7. If $n > 0$ then $\left\lfloor \frac{x}{n}\right\rfloor=\left\lfloor \frac{\lfloor x\rfloor}{n}\right\rfloor.$

| | | | | created: 2019-03-17 05:25:30 | modified: 2019-06-02 08:37:43 | by: bookofproofs | references: [1112], [1272]

1.Proof: (related to "Properties of Floors and Ceilings")