Introduction to Modern Algebra and Number Theory Some of the

[Introduction to Modern Algebra and Number Theory]

Some of the following statements are true and some are false. Prove each true statement and find a counterexample for each false statement. For all real numbers x and y, [x] + [y] = [x + y] - 1.

Solution

rearranging and writing

ceiling fuction(x+y) = ceiling fuction (x) + ceiling fuction(y) + 1

if x and y are integers then

ceiling fuction(x+y) = x + y

if x and y are not integers then

ceiling fuction (x+y) = x + y + ceiling fuction (fractional part of x +fractional part of y )

in worst case (fractional part of x +fractional part of y ) can be equal to 1 hence

ceiling fuction(x+y) = ceiling fuction (x) + ceiling fuction(y) + 1