Question

If solution sets of $P$ and $Q$ are shown by two separate number line

(i) Write down the set-builder form of $P$ and $Q$

(ii) Represent each of the following sets on different number lines: (a) $P\cup Q$, (b) $P\cap Q$, (c) $P\text{'}\cap Q$

Answer 1

Step 1: (i) Evaluate the set-builder form of $P$ and $Q$ from given number lines

$\Rightarrow$ Set-builder form for $P=\{x:-2\le x<6,x\in R\}$

$\Rightarrow$ Set-builder form for $Q=\{x:-6<x<4,x\in R\}$

Step 2: (ii) (a) Evaluate $P\cup Q$ and represent on number line

$\Rightarrow P\cup Q=\{x:-6<x<6,x\in R\}$

(b) Evaluate $P\cap Q$ and represent on number line

$\Rightarrow P\cap Q=\{x:-2\le x<4,x\in R\}$

(c) Evaluate $P\text{'}\cap Q$ and represent on number line

$$\begin{gathered} \Rightarrow P\text{'}\cap Q=Q-P \\ \Rightarrow P\text{'}\cap Q=\{x:-6\le x<-2,x\in R\} \end{gathered}$$

Thus the set-builder form of $P=\{x:-2\le x<6,x\in R\}$ and $Q=\{x:-6<x<4,x\in R\}$