$\xi = \{\text{integers } x : 1 < x < 21\}$ $A = \{\text{integers that are perfect squares}\}$ $B = \{\text{integers that are not prime numbers}\}$ $C = \{\text{integers that are divisible by 5}\}$

1. Identify the elements in the universal set.

Since $\xi=\{\text{integers }x:1<x<21\}$,

$\xi=\{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}.$

2. List the perfect squares in the universal set.

The perfect squares in $\xi$ are

$2^2=4,\quad 3^2=9,\quad 4^2=16.$

So

$A=\{4,9,16\}.$

3. List the non-prime numbers and the multiples of 5 in the universal set.

In $\xi$, the prime numbers are

$2,3,5,7,11,13,17,19.$

Hence the integers in $\xi$ that are not prime are

$B=\{4,6,8,9,10,12,14,15,16,18,20\}.$

The integers in $\xi$ divisible by $5$ are

$C=\{5,10,15,20\}.$

1. Find the elements in $B$ that are not in $C$ and count them.

From part (a),

$B=\{4,6,8,9,10,12,14,15,16,18,20\}$ and $C=\{5,10,15,20\}$.

$B\cap C'$ means elements in $B$ but not in $C$.

So,

$B\cap C'=\{4,6,8,9,12,14,16,18\}$.

Hence, $n(B\cap C')=8$.

1. Use the number of favourable outcomes and divide by the total number of possible outcomes.

From part (b), $n(B\cap C')=8$.

Since $\xi=\{2,3,4,\ldots,20\}$, $n(\xi)=19$.

$P(\text{not prime and not divisible by }5)=\frac{n(B\cap C')}{n(\xi)}=\frac{8}{19}$