$\xi = \{\text{integers } x : 0 < x < 15\}$ $A = \{\text{factors of 15}\}$ $B = \{\text{perfect squares}\}$

1. List the elements in the universal set.

Since $\xi=\{\text{integers }x:0<x<15\}$,
$\xi=\{1,2,3,4,5,6,7,8,9,10,11,12,13,14\}.$

2. Find the elements of sets $A$ and $B$ within the universal set.

$A=\{\text{factors of }15\}$ within $\xi$.

The factors of $15$ are $1,3,5,15$, but $15\notin \xi$.
$A=\{1,3,5\}.$

$B=\{\text{perfect squares}\}$ within $\xi$.
$B=\{1,4,9\}.$

3. Place each element in the correct Venn diagram region.

The common element is
$A\cap B=\{1\}.$

Elements only in $A$ are $\{3,5\}$.

Elements only in $B$ are $\{4,9\}$.

Elements in neither set are
$\{2,6,7,8,10,11,12,13,14\}.$

1. Identify the elements in $A$ that are not in $B$.

From part (a),

$A=\{1,3,5\}$ and $B=\{1,4,9\}$.

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

Since $1$ is in both $A$ and $B$, exclude $1$ from $A$.

Therefore,

$A\cap B'=\{3,5\}$.

1. Identify the relationship between $1$ and $A\cap B$.

From the earlier parts,

$A\cap B=\{1\}$.

Since $1$ is an element of the set $A\cap B$, the correct symbol is $\in$.

Therefore,

$1 \in A\cap B$.