$\xi = \{x:\text{ x is an integer and } 1 \le x \le 10\}$ $A = \{x:\text{ x is a factor of 20}\}$ $B = \{x:\text{ x is a perfect square}\}$ Find

1. Identify and count the elements in set $B$ within the universal set.

Since $\xi=\{1,2,3,4,5,6,7,8,9,10\}$, the perfect squares in $\xi$ are
$B=\{1,4,9\}.$
Hence,
$n(B)=3.$

1. List the relevant elements of $A$ and $B$ within the universal set.

$\xi=\{1,2,3,4,5,6,7,8,9,10\}$

Factors of $20$ in $\xi$ are
$A=\{1,2,4,5,10\}$.

Perfect squares in $\xi$ are
$B=\{1,4,9\}$.

2. Find the elements outside both $A$ and $B$.

$A'\cap B'$ means the elements that are not in $A$ and not in $B$.

So
$A'\cap B'=\{6,7,8,10\}.$

1. Interpret each part of the set expression and combine the regions to be shaded.

$(A \cup B)'$ is the region outside both $A$ and $B$.

$A \cap B$ is the overlapping region of circles $A$ and $B$.

Therefore, for $(A \cup B)' \cup (A \cap B)$, shade:

• the region outside both circles,
• the overlapping region of $A$ and $B$.

Do not shade the $A$-only or $B$-only regions.