Question

Let $U=R$ (the set of all real numbers)
If $A=\{x:x\in R,0<x<2\},$
$B=\{x:x\in R,1<x\le 3\},$
Find $A^{\prime },B^{\prime },A\cup B,A\cap B,A-B.$

Answer 1

We have

$A\prime =R-A=\{x:x\in Randx\in A\}$

$=\{x:(x\in Randx\ge 2)$

or $(x\in Randx\le 0\}$ [See the diagram also]

$=\{x:x\in Randx\ge 2\}\cup \{x:x\in Randx\le 0\}$

$B\prime =\{x:x\in Randx\le 2\}$ $\cup \{x:x\in Randx>3\}$

Similarly $A\cup B=\{x:x\in Rand0<x\le 3\}$

$A\cap B=\{x:x\in Rand1<x<2\}$

$A-B=\{x:x\in Rand0<x\le 1\}$

Note : In such problems use of real number line is useful.