1. Thinking Mathematically
Seventh Edition
Chapter 2
Set Theory
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2. Section 2.4 Set Operations and Venn Diagrams with Three Sets

3. Objectives Perform set operations with three sets.
Use Venn diagrams with three sets.
Use Venn diagrams to prove equality of sets.

4. Example 1: Set Operations with Three Sets
Given
Step 1: Find the complement of C.
Step 2: Find the union of
Step 3: Find the intersection.

5. Venn Diagrams with Three Sets (1 of 2)
The Region Shown in Dark Blue
Region V
This region represents elements that are common to sets A, B and C:
The Regions Shown in Light Blue
Region II
This region represents elements in both sets A and B that are not in set C:
Region IV
This region represents elements in both sets A and C that are not in set B:
Region VI
This region represents elements in both sets B and C that are not in set A:

6. Venn Diagrams with Three Sets (2 of 2)
The Regions Shown in White
Region I
This region represents elements in set A that are in neither sets B nor C:
Region III
This region represents elements in set B that are in neither sets A nor C:
Region VII
This region represents elements in set C that are in neither sets A nor B:
Region VIII
This region represents elements in the universal set U that are not in sets A, B, or C:

7. Example 2: Determining Sets from a Venn Diagram with Three Intersecting Sets
Use the Venn diagram to find:
a. A b. c. d. e.
Solution

8. Example 3: Proving the Equality of Sets (1 of 3)
Prove that
Solution
We can apply deductive reasoning using a Venn diagram to prove this statement is true for all sets A and B. If both sets represent the same regions in this general diagram then this proves that they are equal.

9. Example 3: Proving the Equality of Sets (2 of 3)
Prove that
Solution: Begin with regions
represented by

10. Example 3: Proving the Equality of Sets (3 of 3)
Next, find the regions
Represented by
Since both
are represented
by the same regions, the result proves that they are equal.

11. De Morgan’s Laws
The complement of the intersection of two sets is the union of the complements of those sets.
The complement of the union of two sets is the intersection of the complements of those sets.