Distance to nearest integer

[Sailboat]

I have a question involving {x} which is the distance to the nearest integer such that 0?{x}?1/2. I saw {x} in another thread but my questions are much different.

a. If x is in the real numbers show that {x± 1/2} = 1/2 - {x}

b. with the same condition, show that {x+y} ? {x} + {y} and {x} - {y} ? {x - y}

Much thank you for your consideration.

 

[daon]

For a, you have 2 cases:

Given: 0 <= x <= 1/2
...Case 1: show {x+1/2} = 1/2 - {x}
...Case 2: show {x-1/2} = 1/2 - {x}


I'll do 1)

0 <= x <= 1/2 ===> {x}=x ===> 1/2-{x} = (1-2x)/2

Second,

1/2 <= (x+1/2) <= 1 ===> {x+1/2} = 1 - (x+1/2) = 1/2 - x (why?)

Since {x} = x, we have our result.



For the first part of the second question, we have {x}+{y}=x+y

Case 1: 0 <= x+y <= 1/2

{x+y} = x+y = {x}+{y}

Case 2: 1>= x+y > 1/2

===> 1/2 > x+y-1/2 > 0 ===> {x+y-1/2} = x+y-1/2 = {x}+{y} -1/2 (again, why?)

Hence = {x}+{y} = x+y > 1/2, but {x+y} <= 1/2


Think you can play around and get the others?