Question

Choose the correct relation regarding greatest integer function.

• A. $\left[x\right]-[-x]=2x,x\in Z$
• B. $x-1<\left[x\right]\le x$
• C. $[-[x\left]\right]=-\left[x\right]$
• D. $\left[x\right]\le x<\left[x\right]+1$

Answer 1

Answer: (A), (B), (C), (D)

$\left[x\right]\le x<\left[x\right]+1$

We have studied about some of these properties, which are as follows
$a\left)\right[x]-[-x]=2x,x\in Z$
$b)x-1<[x]\le x$
$c\left)\right[x]\le x<[x]+1$

We have not studied about this property $[-[x\left]\right]=-\left[x\right],$ so let's verify this property.
We will check this through $3$ cases.

Case $1:$ Positive real number $2.5$
$[-[x\left]\right]=-\left[x\right]$
$\Rightarrow [-[2.5\left]\right]=-\left[2.5\right]$
$\Rightarrow [-2]=-2$
$\Rightarrow -2=-2$
We found that both sides are equal, so this is true in case of positive real numbers.

Case $2:$ Negative real number $-2.5$
$[-[x\left]\right]=-\left[x\right]$
$\Rightarrow [-[-2.5\left]\right]=-[-2.5]$
$\Rightarrow [-(-3\left)\right]=-(-3)$
$\Rightarrow 3=3$
We found that both sides are equal, so this is true in case of negative real numbers.

Case $3:$ Integer $3$
$[-[x\left]\right]=-\left[x\right]$
$\Rightarrow [-[3\left]\right]=-\left[3\right]$
$\Rightarrow [-3]=-\left(3\right)$
$\Rightarrow -3=-3$
We found that both sides are equal, so this is true in case of Integers also.

Thus this property is also true for every real number.