$\displaystyle A \cap C$ means the intersection of set A and set C. Intersection means that you want to choose only the numbers that are in BOTH set A and set C. The only numbers that are in common to those two sets are 3 and 9.

$\displaystyle B \cup C$ means the union of set B and set C. Union means to include all the numbers that are in either set B or set C (without listing duplicates twice). The numbers 1, 3, 5, 8, 9, and 11 appear in B and the numbers 2, 3, 5, 6, 9, 12 appear in C. Including all of those numbers in the set symbol gives the correct answer.

Assume Statement 1 alone. $\displaystyle A \cap B \cap C$ is the intersection of the sets $\displaystyle A$, $\displaystyle B$, and $\displaystyle C$. Since Brittany falls in this intersection, she falls in all three sets, and it follows that she likes all three of James Blunt, John Legend, and Pharrell Williams.

Assume Statement 2 alone. $\displaystyle A \cup B \cup C$ is the union of the three sets. Since Brittany falls in this union, she falls in one, two, or all three sets, which means that she likes at least one of James Blunt, John Legend, and Pharrell Williams. However, more information is needed before it can be established whether or not she likes all three.

In set notation, "Jeremy is not a dancer, but he is a singer" can be stated as

$\displaystyle j \in A' \cap B$

That is, Jeremy falls in the intersection of the complement of the set of dancers $\displaystyle A$ and the set of singers $\displaystyle B$.

The opposite of this is that $\displaystyle j$ is in the complement of this set:

$\displaystyle j \in (A' \cap B) '$

By DeMorgan's law, this is equivalent to saying

$\displaystyle j \in (A' ) ' \cup B'$, or

$\displaystyle j \in A \cup B'$.

$\displaystyle j \in A \cap (B' \cup C)$, which is the intersection of $\displaystyle A$ and $\displaystyle B' \cup C$. It follows that $\displaystyle j \in A$ and $\displaystyle j \in B' \cup C$.

$\displaystyle j \in A$, so it follows that Julianna likes Band A. The three choices that state that she does not can be eliminated.

$\displaystyle j \in B' \cup C$. This is the union of $\displaystyle B'$, the complement of $\displaystyle B$ - that is, the set of people not in $\displaystyle B$ - and $\displaystyle C$. It follows that either Julianna does not like Band B, does like Band C, or both. Therefore, it is not true that she likes Band B and does not like Band C. This can be eliminated.

The only possible choice is that Julianna likes Band A and Band C, but not Band B. 

Step 1: Recall the definition for intersection of two sets: The intersection of two sets is a subset that has numbers that is shared between two sets, A and B.

Step 2: Find the shared elements between both sets

Set A and Set B both share 4,11, and 14.

The intersection of both sets is {$\displaystyle 4,11,14$}

Step 1: Define the union of two sets. The union of two sets is all elements that are in two sets, as well as the elements that are shared. 

Step 2: Find the union of the sets:

Set A has elements: $\displaystyle 4,5,6$
Set B has elements: $\displaystyle 4,6,7,8,9,11$

Union of Set A and B ($\displaystyle A \cup B$) is {$\displaystyle 4,5,6,7,8,9,11$}

Step 1: Define what the intersection of two sets means.. The intersection of two sets is defined as a set that has elements that are shared between both original sets..

Step 2: Find any shared elements..

If we look, Set A and Set B do not share any elements..

Therefore, there is no intersection of the two sets.

$\displaystyle B \cap D$

This question asks you to find the intersection of the sets B and D. Include only the numbers that both sets have in common with one another. The only common number with both sets is 4.

$\displaystyle B \cap D=\left \{ 4\right \}$

$\displaystyle (C \cup B) \cap D$

Always start by doing the parentheses first. Do the union of C and B which means to include all the elements from both sets without including duplicates. The numbers do not need to be in any order. Sometimes it can help to put the elements from one entire set before putting the elements in the other

$\displaystyle C \cup B=\left \{ 1, 3, 4, 5, 2, 7, 8, 9, 11, 12\right \}$

Now, take the results from C union B and do the intersection of those elements with set D. Intersection means to include the elements that are common to both sets.

$\displaystyle D=\left \{ 4, 9, 11, 12 \right \}$

The final answer is

$\displaystyle (C \cup B) \cap D=\left \{ 4, 9, 11, 12 \right \}$