$\xi = \{\text{integers } x: 1 \le x \le 16\}$ $V = \{1, 2, 4, 8, 16\}$ $W = \{1, 4, 7, 9, 10, 14\}$ Some of the information is shown on the Venn diagram.

1. Find the elements common to both sets

$V=\{1,2,4,8,16\}$ and $W=\{1,4,7,9,10,14\}$.

The common elements are
$V\cap W=\{1,4\}.$

2. Place the remaining elements in the correct Venn diagram regions

Elements in $V$ only:
$V\setminus W=\{2,8,16\}.$

Elements in $W$ only:
$W\setminus V=\{7,9,10,14\}.$

The elements already outside both circles are
$3,5,6,11,12,13,15.$

1. Recognise the defining property of the elements in $V$

$V=\{1,2,4,8,16\}$

These are exactly the positive factors of $16$.

1. Identify the elements in $W$ that are not in $V$

$V'\cap W$ means elements in $W$ but not in $V$.\n\n$W=\{1,4,7,9,10,14\}$ and $V=\{1,2,4,8,16\}$.\n\nRemove the common elements $1$ and $4$ from $W$.\n\nSo, $V'\cap W=\{7,9,10,14\}$.

1. Identify the elements in $V\cap W'$ and $(V\cup W)'$

$V\cap W'$ means elements in $V$ but not in $W$:
$V\cap W'=\{2,8,16\}.$

$(V\cup W)'$ means elements outside both $V$ and $W$:
$(V\cup W)'=\{3,5,6,11,12,13,15\}.$

2. Form the union and count the number of elements

$=\{2,3,5,6,8,11,12,13,15,16\}.$

There are $10$ elements.

1. Determine the correct set relationship

Since $2\in V$ and $16\in V$, every element of $\{2,16\}$ is in $V$.

Therefore, $\{2,16\}$ is a subset of $V$.

1. Check whether 13 is an element of set $W$ and choose the correct symbol

$W=\{1,4,7,9,10,14\}$

Since $13$ is not listed in $W$,

$13\notin W$