ceiling & floor

logistic_guy

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

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

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

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

Then,

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

It worked in the first case😍
 
Second case.

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

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

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

Then,

3+5.23+5.2\displaystyle \lceil{3 + 5.2}\rceil \neq \lceil 3 \rceil + \lfloor 5.2 \rfloor

It didn't work in the second case😞
 
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