ceiling & floor

[logistic_guy]

For which real numbers $\displaystyle x$ and $\displaystyle y$ is it true that $\displaystyle \lceil{x + y}\rceil = \lceil x \rceil + \lfloor y \rfloor$?

 

[logistic_guy]

First case.

Let $\displaystyle x$ be an integer and let $\displaystyle y$ be an integer.

If $\displaystyle x = 3$ and $\displaystyle y = 5$, then

$\displaystyle \lceil{3 + 5}\rceil = \lceil{8}\rceil = 8$
$\displaystyle \lceil 3 \rceil + \lfloor 5 \rfloor = 3 + 5 = 8$

Then,

$\displaystyle \lceil{3 + 5}\rceil = \lceil 3 \rceil + \lfloor 5 \rfloor$

It worked in the first case

 

[logistic_guy]

Second case.

Let $\displaystyle x$ be an integer and let $\displaystyle y$ be a non-integer.

If $\displaystyle x = 3$ and $\displaystyle y = 5.2$, then

$\displaystyle \lceil{3 + 5.2}\rceil = \lceil{8.2}\rceil = 9$
$\displaystyle \lceil 3 \rceil + \lfloor 5.2 \rfloor = 3 + 5 = 8$

Then,

$\displaystyle \lceil{3 + 5.2}\rceil \neq \lceil 3 \rceil + \lfloor 5.2 \rfloor$

It didn't work in the second case

 

[logistic_guy]

Third case.

Let $\displaystyle x$ be a non-integer and let $\displaystyle y$ be an integer.

If $\displaystyle x = 3.4$ and $\displaystyle y = 5$, then

$\displaystyle \lceil{3.4 + 5}\rceil = \lceil{8.4}\rceil = 9$
$\displaystyle \lceil 3.4 \rceil + \lfloor 5 \rfloor = 4 + 5 = 9$

Then,

$\displaystyle \lceil{3.4 + 5}\rceil = \lceil 3.4 \rceil + \lfloor 5 \rfloor$

It worked in the third case

 

[logistic_guy]

Fourth case.

Let $\displaystyle x$ be a non-integer and let $\displaystyle y$ be a non-integer.

If $\displaystyle x = 3.4$ and $\displaystyle y = 5.2$, then

$\displaystyle \lceil{3.4 + 5.2}\rceil = \lceil{8.6}\rceil = 9$
$\displaystyle \lceil 3.4 \rceil + \lfloor 5.2 \rfloor = 4 + 5 = 9$

Then,

$\displaystyle \lceil{3.4 + 5.2}\rceil = \lceil 3.4 \rceil + \lfloor 5.2 \rfloor$

It worked in the third case

But wait$\displaystyle \textcolor{red}{\bold{!!!}}$

If $\displaystyle x = 3.8$ and $\displaystyle y = 5.3$, then

$\displaystyle \lceil{3.8 + 5.3}\rceil = \lceil{9.1}\rceil = 10$
$\displaystyle \lceil 3.8 \rceil + \lfloor 5.3 \rfloor = 4 + 5 = 9$

Then,

$\displaystyle \lceil{3.8 + 5.3}\rceil \neq \lceil 3.8 \rceil + \lfloor 5.3 \rfloor$

It didn't work in the third case

@fresh_42

What should I say when the fourth case is partially satisfied? And how to solve this problem (the whole problem) by using arbitrary constants?

$\displaystyle \large \textcolor{blue}{\bold{My \ Conclusion}}$
$\displaystyle \lceil{x + y}\rceil = \lceil x \rceil + \lfloor y \rfloor$ when
$\displaystyle \bold{1.} \ x,y \in \mathbb{Z}$
$\displaystyle \bold{2.} \ x \notin \mathbb{Z},y \in \mathbb{Z}$
$\displaystyle \bold{3.} \ x,y \notin \mathbb{Z} \ \text{AND their fractional parts} \leq 1$