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Floor and ceiling functions

Let F(x): R -> Z be the floor function and C(x): R -> Z be the ceiling function. Select which of the following function equations are true (select
zero or more answers).

a. C(x) - F(x) = 1 for all x.
b. C(F(x)) = F(x) for all x
c. C(F(x)) = F(C(x)) - 1 for all x
d. F(F(x)) = F(x) for all x

Solution Preview

a. False.
Counter example: C(1) - F(1) = 1 - 1 = 0.
b. True
If x is an integer, then C(F(x)) = C(x) = x = F(x).
If x is not an integer, then x between two integers ...

Solution Summary

This looks at a floor and ceiling function, and then compares the two functions (equality and other statements).