Question

Show that [x,y]isO(xy)

Short answer

Hence, [x,y] is O(xy)

Step-by-step solution

Step 1:

Let $f(x,y)=\left[xy\right]$and

Let $k_1=k_2=1$

We know $x>k_1$and$y>k_2$

$\Rightarrow x>1$and y > 1

Step 2:

Applying Big-O notation definition we get

$\left|f\right(x,y\left)\right|=\left[xy\right]$

We know that $\left|a\right|\le a$for any value of {Property of floor function}

$$\begin{gathered} \Rightarrow \left|f\right(x,y\left)\right|\le xy \\ \Rightarrow \left|f\right(x,y\left)\right|\le \left|xy\right| \end{gathered}$$

Comparing the inequality with role="math" localid="1668531415370" $\left|f\right(x,y\left)\right|\le C\left|g\right(x,y\left)\right|$we get C = 1 and

[x,y] is O(xy)