Question

# Under what conditions is \lfloor x\rfloor=\lceil x\rceil-1

Discrete math
Under what conditions is $$\displaystyle\lfloor{x}\rfloor=\lceil{x}\rceil-{1}$$

2021-08-04
Step 1
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,
$$\displaystyle{R}-{Z}$$