Question

Which of the following is true about $\left[x\right]$ i.e greatest integer function?

• A. $x-1<\left[x\right]\le x$
• B. $[-x]=-1-\left[x\right]$, when x is not an integer
• C. $\left[\right[x\left]\right]=\left[x\right]$
• D. $[x+n]=\left[x\right]+n,n\in I$

Answer 1

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

$[x+n]=\left[x\right]+n,n\in I$

(A) $\left[x\right]$ is an integer say n,
$\left[\left[x\right]\right]=\left[n\right]=n$, because $n$ is an integer.

(B) Since for all $x\in I$
$[x+n]=\left[x\right]+n$.
This is true.

(C) $x-1<\left[x\right]\le x$

$\Rightarrow x=\left[x\right]+\left\{x\right\}$

$\Rightarrow \left[x\right]=x-\left\{x\right\}$
$\because \left\{x\right\}\in [0,1)$

$\Rightarrow \left[x\right]$ is less than $x$ and it will be equal to $x$ when $\left\{x\right\}$ is zero.
$\left\{x\right\}$ is less then $1$.
So $\left[x\right]=x-\left\{x\right\}$, will be greater than $x-1$ .

(D) We know that
$\left[x\right]+[-x]=\{\begin{array}{cc}0,& ifx\in Z\\ -1,& ifx\notin Z\end{array}$
So, This justify statement (D).