$\varepsilon = \{\text{integer } x : 0 < x \le 12\}$ $A = \{\text{prime numbers}\}$ $B = \{\text{numbers that have at least 2 distinct factors}\}$

1. Show that all elements of $A$ are also elements of $B$.

A prime number has exactly two distinct factors, namely $1$ and itself. Therefore every prime number has at least $2$ distinct factors, so every element of $A$ is in $B$. Hence $A \subseteq B$.

2. Show that $A$ is not equal to $B$, so the subset is proper.

There are numbers in $B$ that are not prime, for example $4$ has at least $2$ distinct factors but is not prime. Hence $B$ contains at least one element not in $A$, so $A \ne B$ and $A$ is a proper subset of $B$.

1. Write the universal set and identify the elements of $A$.

The universal set is
$\varepsilon=\{1,2,3,4,5,6,7,8,9,10,11,12\}.$
The prime numbers in this set are
$A=\{2,3,5,7,11\}.$

2. Place elements correctly in the Venn diagram according to $A \subset B$.

Since $A$ is a proper subset of $B$, draw $A$ entirely inside $B$.

Numbers in $B$ but not in $A$ have at least $2$ distinct factors but are not prime:
$B\setminus A=\{4,6,8,9,10,12\}.$
The only element not in $B$ is
$\varepsilon\setminus B=\{1\}.$
So place $2,3,5,7,11$ inside $A$; place $4,6,8,9,10,12$ inside $B$ but outside $A$; place $1$ outside $B$ but inside $\varepsilon$.