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Q.
Let $A = \left\{x: x \in R, | x | < 2\right\}$, $B =\left\{ x : x \in R, |x - 2| \ge 2\right\}$ and $A \cup B = R- C$, then the set $C$ equals
Sets
Solution:
Given : $A = \left\{x: x \in R, | x | < 2\right\}$
and $B =\left\{ x : x \in R, |x - 2| \ge 2\right\}$
$\Rightarrow A = -2 < x < 2$
and $B =\left\{ x : x \in R, x - 2 \le - 2 \,\text{or} \,x - 2 \ge 2\right\}$
$= \left\{x : x \in R, x \le 0 \,\text{or}\, x \ge 4\right\}$
$\Rightarrow A = ]-2, 2[\,\&\, B = (-\infty, 0] \cup [4, \infty)$
Now $A \cup B = (-\infty, 2) \cup [4, \infty) = R - [2, 4)$
$\therefore C = [2, 4)$ i.e. $2 \le x < 4$
