## 2.3 Outline

- Combined operations with sets
- order of operations
- Venn diagrams

- De Morgan’s laws
- Venn diagrams with three sets
- Survey problems

## 2.3 Essential Ideas

**Combined Operations**

For combined operations, perform operations in parenthesis first, then proceed from left to right. To find a complement, first find the set under the overbar, and then shade all parts not contained under the overbar.

**De Morgan’s Laws**

There are two parts to De Morgan’s Law.

The complement of a union is the intersection of the complements.

The complement of an intersection is the union of the complements.

**Survey Problems**

Draw intersecting circles to represent the different categories (sets) in the

survey.

Next, try to determine the number of elements in the intersection of all of the

sets.

Fill in the other inner portions.

Work from the inner portions to the outer portions, being sure to subtract the

numbers in the portions already accounted for; move systematically from one region to another.

Finally, answer the questions asked by reading from the Venn diagram.