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
\(\displaystyle \bold{1.} \ x,y \in \mathbb{Z}\)
\(\displaystyle \bold{2.} \ x \notin \mathbb{Z},y \in \mathbb{Z}\)
\(\displaystyle \bold{3.} \ x,y \notin \mathbb{Z} \ \text{AND their fractional parts} \leq 1 \)