Question

Choose the wrong relation where $\left[\right]$ denotes the greatest integer function and $n\in Z.$

• A. $\left[x\right]>n\iff x\ge n+1.$
• B. $\left[x\right]\le n\iff x<n+1$
• C. $\left[x\right]\ge n\iff x>n+1$
• D. $\left[x\right]=\left[\frac{x}{2}\right]+\left[\frac{x+1}{2}\right]$

Answer 1

The correct option is C $\left[x\right]\ge n\iff x>n+1$

The properties of greatest integer function are as follows:

1. $\left[x\right]\ge n\iff x\ge n$
2. $\left[x\right]>n\iff x\ge n+1.$
3. $\left[x\right]\le n\iff x<n+1$
4. $\left[x\right]<n\iff x<n$
5. $\left[x\right]=\left[\frac{x}{2}\right]+\left[\frac{x+1}{2}\right]$
6. $\left[x\right]+\left[y\right]\le [x+y]\le \left[x\right]+\left[y\right]+1$

Hence, the relation in option (d) is not satisfied.