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

1. Identify the elements of $B$ within the universal set.

The universal set is $\xi=\{1,2,3,4,5,6,7,8,9,10\}$.

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

2. Count the number of elements in $B$.

There are $3$ elements in $B$.

$n(B)=3$

1. List the relevant sets within the universal set.

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

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

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

2. Find the elements not in both $A$ and $B$.

$A'=\{3,6,7,8,9\}$

$B'=\{2,3,5,6,7,8,10\}$

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

1. Interpret each part of the set expression as regions on the Venn diagram.

$(A \cup B)'$ means the region outside $A \cup B$, so it is the region outside both circles.

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

2. Combine the two required regions using union.

$(A \cup B)' \cup (A \cap B)$ means shade both:

• the region outside both circles,
• the overlap $A \cap B$.

Do not shade the parts in $A$ only or $B$ only.