Question

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

Answer 1

$y=\frac{1}{x}\Rightarrow xy=1$.
$\therefore$ $A$ is the set of all points on the rectangular hyperbola. $xy=1$ with branches in I and III quadrants, $y=-x$ represents a line with slope ${}^{\prime }-1^{\prime }$ and $c$ equal to ${}^{\prime }{0}^{\prime }$.
$\therefore$ $B$ is the set of all points on this line. Since the graphs of $xy=1$ and $y=-x$ are non-intersecting, we have $A\cap B=\varphi$.