**A Set**

A **set** is a group of elements.

For example, set “A” could be a group of four elements:

A = {3, 9, 12, 15}

And set “B” could also be a group of four elements:

B = {3, 4, 12, 19}

**Union**

The **union** of two sets, denoted as “∪”, is simply the list of elements that are in set A *or *set B. For example, the union of set A and B would be written as:

A∪B = {3, 4, 9, 12, 15, 19}

**Intersection**

The **intersection** of two sets, denoted as “∩”, is the list of elements that are in both set A *and *B. The intersection of set A and set B would only include the two elements that are in both sets:

A = {**3**, 9, **12**, 15}

B = {**3**, 4, **12**, 19}

A∩B = {3, 12}

**Relative Complement**

The **relative complement** of two sets is simply the difference between two sets.

For example, the “relative complement of B in A” gives us all of the elements in A that are *not *in B. We denote this as A \ B.

To find A \ B, we start with all of the elements in A and simply *remove *any elements from A that are also in B:

A = {**3, 9, 12, 15**}

B = {3, 4, 12, 19}

A \ B = {**9, 15**}

We could also find the “relative complement of A in B”, which gives us all of the elements in B that are *not *in A. We denote this as B \ A:

B = {**3, 4, 12, 19**}

A = {3, 9, 12, 15}

B \ A = {**4, 19**}

**Universal Set**

A **universal set** is a set that contains all of the possible elements of something we care about. For example, suppose we have set “U”, which is our universal set:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Suppose we also have set “A”:

A = {1, 2, 3}

**Absolute Complement**

The **absolute complement** of A, denoted as A’, is all of the elements in the universal set that are *not *in A:

U = {**1, 2, 3, 4, 5, 6, 7, 8, 9, 10**}

A = {1, 2, 3}

A’ = {**4, 5, 6, 7, 8, 9, 10}**

**Subsets, Null Sets, and Equal Sets**

If every element in set A is also in set B, then set A is a **subset** of set B. Suppose A = {1, 2, 3} and B = {1, 2, 3, 4, 5}. In this case, set A is a subset of set B because every element in A is also in B.

A set that contains no elements is called a **null set** or an **empty set, **and is usually written as {}. Suppose A = {1, 2, 3} and B = {4, 5, 6}. Recall that the *intersection* of these two sets is the set of numbers that are in *both *A and B. In this case, they share no common elements so the intersection would be {}, an empty set.

Two sets are **equal** if they share all of the same elements. For example, if set A = {14, 8, 2} and set B = {14, 8, 2}, then sets A and B are equal.