Step 1: Recall the definition of a subset. A subset is a small part of a bigger given set; ALL numbers in the subset must be in the original set.

Step 2: We look at the original set and then look at the answers. If there is any answer that has an element that is not showing up in the original set, than that is the right answer.

Let's look at: . The  is in the original set, but the  is not. Since we have a number in the subset that is not in the original set, we can say that  is not a subset.

The other two answers with brackets are subsets of the original set. All elements in the subsets are present in the original set.

The empty set is always a subset of any given set. 

The answer is 

Step 1: Define a subset. A subset of a bigger set is a smaller part where all the elements in the subset must be present in the bigger set.

 The answer is . All numbers in this subset are in the bigger set, which is provided in the question...

Step 1: Define a subset

A subset of a big set is a set that has some of the elements that are in the bigger parent set.

The answer is  because both  and  are both in the bigger set.

Step 1: Determine the difference between Infinite and Finite Sets...

Finite Sets: A set that has very limited elements in it
Infinite Sets: Any set where I can always find another number between two given boundaries.

The Set of integers and rational numbers are infinite sets..
All numbers between  and  is also an infinite set because I can come up with infinite decimal numbers..

The set of whole numbers between  and  inclusive is a finite set because I ask for a specific type of number.. a whole number. Whole numbers cannot be decimals..

Note: The set of real numbers , natural numbers  are both infinite sets.
Step 1: The set of real numbers between two numbers  is also an infinite set.

Step 2: The set of whole numbers between  and  inclusive is finite because there are only three numbers that are represented by the set. These numbers are .

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.

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.  is the intersection of the sets , , and . 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.  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

That is, Jeremy falls in the intersection of the complement of the set of dancers  and the set of singers .

The opposite of this is that is in the complement of this set:

By DeMorgan's law, this is equivalent to saying

, or

.

, which is the intersection of  and . It follows that  and .

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

. This is the union of , the complement of  - that is, the set of people not in  - and . 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.