$\xi = \{x : x \text{ is an integer},\; 4 \le x \le 16\}$ $A = \{4, 9, 16\}$ $B = \{4, 6, 7, 8, 9, 10, 16\}$

1. Recognise the relationship between sets $A$ and $B$ and draw the correct Venn diagram structure.

Since
$A=\{4,9,16\}$
and every element of $A$ is also in
$B=\{4,6,7,8,9,10,16\},$
we have $A\subset B$.

So draw a rectangle for $\xi$, a circle $B$ inside the rectangle, and a smaller circle $A$ completely inside circle $B$.

2. Place every element of the universal set in the correct region.

The elements in $A$ are
$4,9,16.$

The elements in $B$ but not in $A$ are
$6,7,8,10.$

The elements in $\xi$ but not in $B$ are
$5,11,12,13,14,15.$

1. Recognise the elements of set $A$ and describe their common property.

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

These are square numbers:
$4=2^2,\quad 9=3^2,\quad 16=4^2.$

1. Identify that all elements of $A$ are also in $B$, so no element of $A$ is outside $B$.

$A = \{4,9,16\}$ and $B = \{4,6,7,8,9,10,16\}$.

Since $4,9,16$ are all in $B$, there are no elements that are in $A$ but not in $B$.

Therefore,
$A \cap B' = \varnothing.$