Question

If $A=\left\{\right(x,y):y=e^x,x\in R\},B=\left\{\right(x,y):y=x,x\in R\}$, then

• A. $B\subset A$
• B. $A\subset B$
• C. $A\cap B=\varphi$
• D. $A\cup B=A$

Answer 1

Answer: (C)

$A\cap B=\varphi$

Let us take a function $g\left(x\right)=e^x-x$
$g^{\prime }\left(x\right)=e^x-1$
Thus, $g^{\prime }\left(0\right)=0$ and $g^{\prime }\left(x\right)>0$ for $x>0$
Hence, $g\left(x\right)$ is an increasing function for $x>0$.
Now, for $x<0,e^x$ is positive and $x$ is negative, so the two graphs won't intersect in that domain.
For $x>0,g\left(x\right)$ being an increasing function, the difference between $e^x$ and $x$ goes on increasing.
Hence the two functions never intersect.
$A\cap B=\varphi$