### Introduction to Sets

A set is simply a collection of objects or elements. For example, A is set of 5 numerical members as represented below:

*A = {2, 3, 5, 7, 9}*

### Overlapping Sets

Two or more sets which contain one or more common members are overlapping sets. There are some tricky word problems which involve use of 2 or 3 overlapping sets. Things are really simple when only one set is considered. However, as the number of sets increases, it becomes more comples and interesting too.

### Venn diagrams

Venn diagrams are the best method for solving problems with overlapping sets. If there are two overlapping sets, you need a** two-circle Venn diagram**:

This venn diagram contains four discrete regions:

- A = those elements in just the left circle
- B = those element in both categories, in the overlapping region
- C = those elements in just the right circle
- Those elements that are not members of any set

If there are three overlapping categories, we use a **three-circle Venn diagram**:

This venn diagram has eight discrete regions:

- A = members of all three circles
- B = members of the green and blue circles, but not the red circle
- C = members of the green and red circles, but not the blue circle
- D = members of the blue and red circles, but not the green circle
- E = members of the green circle but of neither the blue nor the red circles
- F = members of the blue circle but of neither the green nor the red circles
- G = members of the red circle but of neither the green nor the blue circles
- Those elements that are not members of any of the three sets

### Example Questions

**Example 1: Each of 25 people is enrolled in history, mathematics, or both. If 20 are enrolled in history and 18 are enrolled in mathematics, how many are enrolled in both history and mathematics?**

The 25 people can be divided into three sets: those who study history only, those who study mathematics only, and those who study history and mathematics.

If n is the number of people enrolled in both courses, 20-n is the number enrolled in history only, and 18-n is the number enrolled in mathematics only.

Since there is a total of 25 people,(20-n)+n+(18-n) = 25, or n=13. Thirteen people are enrolled in both history and mathematics.