Question

If sets $A$ and $B$ are defined as $A=\left\{(x,y):y=\frac{1}{x},0\equiv x\in R\right\}$ and $B=\left\{\right(x,y):y=-x,x\in R\}$, then

• A. $A\cap B=A$
• B. $A\cap B=B$
• C. $A\cap B=\varnothing$
• D. None of these

Answer 1

The correct option is C

$A\cap B=\varnothing$

Explanation for the correct option:

Step 1: Draw the required diagram

Step 2: Finding the relation between set $A$ and set $B$ :

$$\begin{gathered} \mathrm{y}=\frac{1}{\mathrm{x}} \\ \Rightarrow \mathrm{xy}=1 \end{gathered}$$
So, $A$ is the set of all points on the rectangular hyperbola.

$xy=1$ with branches in I and III quadrants.

Whereas, $y=-x$ represents a line with slope $-1$and $c=0$.
$\therefore B$ is the set of all points on this line.

Since the graphs of $xy=1$ and $y=-x$ are non-intersecting,

We have $\mathrm{A}\cap \mathrm{B}=\varphi$.

Hence, option C is correct.