Question

Let $A=\{x:x\in R,|x|<1\}$;
$B=\{x:x\in R,|x-1|\ge 1\}$ and

$A\cup B=R-D$, then the set $D$ is

• A. $\{x:1<x\le 2\}$
• B. $\{x:1\le x<2\}$
• C. $\{x:1\le x\le 2\}$
• D. None of these

Answer 1

The correct option is B

$\{x:1\le x<2\}$

Since $A=\{x:x\in R,|x|<1\}$
$\Rightarrow A=\{x:x\in R,-1<x<1\}$

and $B=\{x:x\in R,|x-1|\ge 1\}$
$\Rightarrow B=\{x:x\in R,x-1\le -1\text{or}x-1\ge 1\}$

$=\{x:x\in R,x\le 0\text{or}x\ge 2\}$

Now, $A\cup B=\{x:x<1\text{or}x\ge 2\}$

It is given that $A\cup B=R-D$,
using the above information, we can conclude that $D=\{x:x\in R,1\le x<2\}$.