Let’s dive into some key concepts we will use throughout this lesson: *union*, *intersection*, and *complement*.

#### Union

The *union* of two sets encompasses any element that exists in either one or both of them. We can represent this visually as a *Venn diagram*.

For example, let’s say we have two sets, *A* and *B*. *A* represents rolling an odd number with a six-sided die (the set *{1, 3, 5}*). *B* represents rolling a number greater than two (the set *{3, 4, 5, 6})*. The union of these two sets would be everything in either set *A*, set *B*, or both: *{1, 3, 4, 5, 6}*. We can write the union of two events mathematically as *(A or B)*.

#### Intersection

The *intersection* of two sets encompasses any element that exists in both of the sets. Visually:

The intersection of the above sets (*A* represents rolling an odd number on a six-sided die and *B* represents rolling a number greater than two) includes any value that appears in both sets: *{3, 5}*. We can write the intersection of two events mathematically as *(A and B)*.

#### Complement

Lastly, the *complement* of a set consists of all possible outcomes outside of the set. Visually:

Consider set *A* from the above example (rolling an odd number on a 6-sided die). The complement of this set would be rolling an even number: *{2, 4, 6}*. We can write the complement of set *A* as *A ^{C}*. One key feature of complements is that a set and its complement cover the entire sample space. In this die roll example, the set of even numbers and odd numbers would cover all possible rolls:

*{1, 2, 3, 4, 5, 6}*.

### Instructions

To the right is an applet that shows two overlapping circles: *A* and *B*. The two dropdown menus allow you to choose from different events. Event *A* can be any of the following:

- The outcome of a die roll is even
- The outcome of a die roll is a five
- The outcome of a die roll is less than or equal to three

Event *B* can be any of the following:

- The outcome of a die roll is odd
- The outcome of a die roll is anything but five
- The outcome of a die roll is greater than or equal to two

Using the applet, find the intersection and union between the following two events:

- The outcome of a die roll is less than or equal to three
- The outcome of a die roll is anything but five

Using the applet, find the complement of the following events:

- The outcome of a die roll is greater than or equal to two
- The outcome of a die roll is five

Feel free to play around with it more!