First define floor and ceiling function.

Floor function is defined as a function that takes an input x and gives the output the largest integer less than or equal to x.

Ceiling function is defined as a function that takes an input x and gives the output the smallest integer greater than or equal to x.

Now define using an example of both functions.

example of floor function:

\(\displaystyle\lfloor{2.3}\rfloor={2}\)

example of ceiling function:

\(\displaystyle\lceil{2.3}\rceil={3}\)

Step 2

Now define where the function is true or false.

First consider the case of integers.

let \(\displaystyle{x}={3}\)

\(\displaystyle\lfloor{3}\rfloor={3}\)

\(\displaystyle\lceil{3}\rceil={3}\)

but \(\displaystyle{3}\ne{q}{3}-{1}\) that is \(\displaystyle{3}\ne{q}{2}\)

so, this statement is not true for integers.

Step 3

Now consider the case of real numbers other than integers.

First take case of the positive real fraction numbers.

let \(\displaystyle{x}={2.3}\)

\(\displaystyle\lfloor{2.3}\rfloor={2}\)

\(\displaystyle\lceil{2.3}\rceil={3}\)

but \(\displaystyle{2}={3}-{1}\) that is \(\displaystyle{2}={2}\)

so, this statement is true for positive real fraction.

Step 4

Now, take case of negative real fraction numbers.

let \(\displaystyle{x}=-{2.3}\)

\(\displaystyle\lfloor-{2.3}\rfloor=-{3}\)

\(\displaystyle\lceil-{2.3}\rceil=-{2}\)

but \(\displaystyle-{3}=-{2}-{1}\) that is \(\displaystyle-{3}=-{3}\)

so, this statement is true for negative real fraction.

Step 5

So, the statement is true for all real numbers except integers. That is,

\(\displaystyle{R}-{Z}\)