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# Introduction

## What you’ll learn to do: Use proper set notation and Venn diagrams to depict set relations and operations

It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets to understand relationships between groups, and to analyze survey data.

### Learning Outcomes

• Describe memberships of sets and relationships between sets while using correct set notation.
• Perform the operations of union, intersection, and complement, on sets in the context of Venn diagrams.
• Be able to draw and interpret Venn diagrams of set relations and operations.
• Solve real-life problems using Venn Diagrams.

# Venn Diagrams

An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of distinct items, or elements, can form a set.  To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations are now called Venn Diagrams.   In a Venn diagram, we use circles to represent the sets described in our scenario as seen in Figure 1 below. We place these circles inside of a box, which represents the "universe" we are considering for the particular scenario and are filled in with their respective elements.  Overlapping areas indicate regions that are common between the different sets.  Regions outside of a circle represent elements that are not in that set.  Basic Venn diagrams can be used to visualize the relationships between up to three distinct sets.

Figure 1: Venn Diagram

# Universal Set and Complement

The universal set is the set that contains all the elements we are interested in for our particular scenario. This set would be defined by the context. We denote the universal set (or universe) with the notation $U$. In a Venn diagram, the universal set is represented by the box that contains all of the circles.  In Figure 1 above, the universal set is the set of all plants. The complement of a set contains everything that is not in that particular set. For a set $A$, we denote the complement with the notation $A^c$.  The complement of a set is relative to the universal set, so  $A^c$ contains all the elements in the universal set that are not in $A$. In a Venn diagram, a complement is represented by all of the regions that are not inside of the circle representing the set as shown in Figure 2 below.

Figure 2: Complement of $A$

### Complement

A universal set is a set that contains all the elements we are interested in and is defined by the context of our scenario. The complement of a set A contains everything that is not in the set A. We denote the complement with the notation AcThe complement of a set is relative to the universal set, so Ac contains all the elements in the universal set that are not in A.

### a new use for a superscript

Notice in the descriptions of the notation introduced above that the complement of a set is denoted $A^{c}$. This superscript is not an exponent. It is a decoration that denotes the complement of a set.

### Example

1. If we were discussing searching for books, the universal set might be all the books in the library.
2. If you were working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers.
3. If we were grouping your Facebook friends, the universal set might be all your Facebook friends.  If we consider a set, $A,$ of your relatives who are Facebook friends, then $A^{c}$ would be the set of your Facebook friends to whom you are not related.

# Union and Intersection of Sets

Commonly, sets interact with each other. The union of two sets includes all the elements that are contained within either set (or both sets). For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets. The union is notated A B. Figure 3 illustrates the Venn diagram representation of the union of two sets. Notice that all of the regions enclosed by circle A and circle B are shaded.

Figure 3: Union of Two Sets

Or, perhaps you decide to only invite the friend you have in common.  This type of set interaction is called the intersection. The intersection of two sets contains only the elements that are in both sets. The intersection is notated $A\cap B$. Figure 4 illustrates the Venn diagram representation of the intersection of two sets. Notice that only the overlap of circle A and circle B is shaded.

Figure 4: Intersection of Two Sets

### Union and Intersection

The union of two sets contains all the elements contained in either set (or both sets). The union is notated A B.  The intersection of two sets contains only the elements that are in both sets. The intersection is notated A B.

### union and intersection symbols

The intersection $\cap$ and union $\cup$ symbols look a little like letters in the alphabet. In fact, that's a trick for remembering them. The union symbol looks like  a capital U, for union. The intersection symbol looks a little like a big lower-case n, for in-tersect

### Examples

Example 1: Create a Venn diagram to visualize the set: Ac B

Answer:   A contains all elements that are not in the set A.  Ac B will contain the elements in set B that are not in set A.

Example 2:  Use a Venn diagram to visualize the set: (H F)c W.

Answer: We’ll start by identifying everything in the set H F Now, (H F)c ⋂ W will contain everything in the set that is not in the set identified above.