$\xi = \{\text{integers } x: 2 < x \le 12\}$ $A = \{\text{prime numbers}\}$ $B = \{\text{factors of 12}\}$ $C = \{\text{greater than } \sqrt{100}\}$ List the elements in $A \cap C'$ , Answer ... $A \cup B$ . Answer ... It is given that $M \cap N = \emptyset$ and $L \subset N$. Complete and label the Venn diagram below for the sets L, M and N.

1. Identify the relevant sets within the universal set

The universal set is
$\xi=\{3,4,5,6,7,8,9,10,11,12\}$.

$A=\{\text{prime numbers in }\xi\}=\{3,5,7,11\}$.

$C=\{\text{numbers in }\xi\text{ greater than }\sqrt{100}\}$.
Since $\sqrt{100}=10$,
$C=\{11,12\}$.

2. Find the complement of $C$ and intersect it with $A$

$C'=\{3,4,5,6,7,8,9,10\}$.

So
$A\cap C'=\{3,5,7,11\}\cap\{3,4,5,6,7,8,9,10\}=\{3,5,7\}.$

1. List the elements of $A$ and $B$ within the universal set

The universal set is $\xi=\{3,4,5,6,7,8,9,10,11,12\}$.

Prime numbers in $\xi$:
$A=\{3,5,7,11\}$

Factors of $12$ in $\xi$:
$B=\{3,4,6,12\}$

2. Find the union of $A$ and $B$

$A\cup B$ contains all elements that are in $A$ or in $B$.

$A\cup B=\{3,4,5,6,7,11,12\}$

1. Represent $M \cap N = \emptyset$ on the Venn diagram

$M \cap N = \emptyset$ means that $M$ and $N$ have no common elements.

So draw the circles for $M$ and $N$ separately, with no overlap.

2. Represent $L \subset N$ on the Venn diagram

$L \subset N$ means every element of $L$ is also in $N$.

So draw the circle for $L$ completely inside the circle for $N$, and label all three circles $L$, $M$ and $N$.