$\xi = \{x : x \text{ is an integer and } -3 \le x < 7\}$ $P = \{x : -3 < x < 3\}$ $Q = \{x : 0 < x \le 3\}$

1. List the elements of the universal set and the two subsets.

Since $x$ is an integer and $-3 \le x < 7$,
$\xi=\{-3,-2,-1,0,1,2,3,4,5,6\}.$

For $P$, $-3<x<3$, so
$P=\{-2,-1,0,1,2\}.$

For $Q$, $0<x\le3$, so
$Q=\{1,2,3\}.$

2. Identify the elements in each Venn diagram region.

The common elements are
$P\cap Q=\{1,2\}.$

So the regions are:
$P\text{ only}=\{-2,-1,0\},$
$Q\text{ only}=\{3\},$
$\xi\text{ outside both }P\text{ and }Q=\{-3,4,5,6\}.$

3. Draw and label the Venn diagram accurately.

Draw a rectangle labelled $\xi$ containing two overlapping circles labelled $P$ and $Q$.

Place:

• $1,2$ in the overlap,
• $-2,-1,0$ in $P$ only,
• $3$ in $Q$ only,
• $-3,4,5,6$ outside both circles but inside $\xi$.

1. List the elements in the universal set and remove the elements of $P$.

$\xi = \{-3,-2,-1,0,1,2,3,4,5,6\}$

$P = \{-2,-1,0,1,2\}$

So the elements not in $P$ are

$P' = \{-3,3,4,5,6\}$.

1. Find the common elements of $P$ and $Q$.

From the earlier listing,

$P = \{-2,-1,0,1,2\}$ and $Q = \{1,2,3\}$.

The elements in both sets are $1$ and $2$.

So, $P \cap Q = \{1,2\}$.

1. Find the union of $P'$ and $Q$, then count its elements.

From earlier parts,

$P'=\{-3,3,4,5,6\}$ and $Q=\{1,2,3\}$.

So

$P'\cup Q=\{-3,1,2,3,4,5,6\}$.

Hence

$n(P'\cup Q)=6$.