1. THE NATURE OF SETS Copyright © Cengage Learning. All rights reserved. 2

2. 2.3 Applications of Sets

3. 3 Combined Operations with Sets

4. 4 Example 1 – Order of operations Verbalize the correct order of operations and then illustrate the combined set operations using Venn diagrams: Solution: a. This is a combined operation that should be read from left to right. First find the complements of A and B and then find the union. This is called a union of complements.

5. 5 Example 1 – Solution Step 1 Shade A (vertical lines), then shade B (horizontal lines). cont’d

6. 6 Example 1 – Solution Step 2 is every portion that is shaded with horizontal or vertical lines. We show that here using a color highlighter. cont’d

7. 7 Example 1 – Solution b. This is a combined operation that should be interpreted to mean which is the complement of the union. First find A  B (vertical lines), and then find the complement (color highlighter). This is called the complement of a union. We show only the final result. cont’d

8. 8 De Morgan’s Laws

9. 9 Notice from Example 1 that If they were equal, the final highlighted color portions of the Venn diagrams would be the same. The next example takes us a step further by showing what does equal.

10. 10 Example 2 – De Morgan’s Law Prove Solution: We use Pólya’s problem-solving guidelines for this example. Understand the Problem. We wish to prove the given statement is true for all sets A and B, so we cannot work with a particular example. Devise a Plan. The procedure is to draw separate Venn diagrams for the left and the right sides, and then to compare them to see if they are identical.

11. 11 Example 2 – Solution Carry Out the Plan. Step 1 Draw a diagram for the expression on the left side of the equal sign, namely. The final result is shown with color highlighter. cont’d

12. 12 Example 2 – Solution Step 2 Draw a diagram for the expression on the right side of the equal sign. First draw A (vertical lines) and then B (horizontal lines). The final result, is the part with both vertical and horizontal lines (as shown with the color highlighter) in the right portion of Figure 2.13. cont’d Figure 2.13 De Morgan’s Law

13. 13 Example 2 – Solution Step 3 Compare the portions shaded by the color highlighter in the two Venn diagrams. Look Back. They are the same, so we have proved cont’d

14. 14 De Morgan’s Laws The result proved in Example 2 is called De Morgan’s law for sets.

15. 15 Venn Diagrams with Three Sets

16. 16 Example 4 – Union is associative If (P  Q)  R = P  (Q  R), we say that the operation of union is associative. Is the operation of union for sets an associative operation? Solution: We use Pólya’s problem-solving guidelines for this example. Understand the Problem. Even though we cannot answer this question by using a particular example, we can use one to help us understand the question.

17. 17 Example 4 – Solution Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, P = {1, 4, 7}, Q = {2, 4, 9, 10}, and R = {6, 7, 8, 9}. Is the following true? (P  Q)  R = P  (Q  R)? (P  Q)  R = {1, 2, 4, 7, 9, 10}  {6, 7, 8, 9} = {1, 2, 4, 6, 7, 8, 9, 10} (P  Q)  R = {1, 4, 7}  {2, 4, 6, 7, 8, 9, 10} = {1, 2, 4, 6, 7, 8, 9, 10} cont’d

18. 18 Example 4 – Solution For this example, the operation of union for sets is associative. If we had observed (P  Q)  R  P  (Q  R), then we would have had a counterexample. Although they are equal in this example, we cannot say that the property is true for all possibilities. However, all is not lost because it did help us to understand the question. Devise a Plan. Use Venn diagrams. cont’d

19. 19 Example 4 – Solution Carry Out the Plan. Recall that the union is the entire shaded area. Look Back. The operation of union for sets is an associative operation since the parts shaded in yellow are the same for both diagrams. cont’d

20. 20 Survey Problems

21. 21 Survey Problems There is a formula for the number of elements, but it is easier to use Venn diagrams, as illustrated by Example 5. The usual procedure is to fill in the number in the innermost region first and work your way outward through the Venn diagram using subtraction.

22. 22 Example 5 – Survey problem A survey of 100 randomly selected students gave the following information. 45 students are taking mathematics. 41 students are taking English. 40 students are taking history. 15 students are taking math and English. 18 students are taking math and history. 17 students are taking English and history. 7 students are taking all three.

23. 23 Example 5 – Survey problem a. How many are taking only mathematics? b. How many are taking only English? c. How many are taking only history? d. How many are not taking any of these courses? Solution: We use Pólya’s problem-solving guidelines for this example. Understand the Problem. We are considering students who are members of one or more of three sets. If U represents the universe, then | U | = 100.

24. 24 Example 5 – Solution We also define the three sets: M = {students taking mathematics}, E = {students taking English}, H = {students taking history}. Devise a Plan. The plan is to draw a Venn diagram, and then to fill in the various regions. We fill in the innermost region first, and then work our way outward (using subtraction) until the numbers of elements of the eight regions formed by the three sets are known. cont’d

25. 25 Example 5 – Solution Carry Out the Plan. Step 1 We note |M  E  H| = 7 in Figure 2.16a. cont’d Figure 2.16a First step is inner region

26. 26 Example 5 – Solution Step 2 Fill in the other inner portions. |E  H| = 17, but 7 have previously been accounted for, so an additional 10 members (17 – 7 = 10) are added to the Venn diagram (see Figure 2.16b). |M  H| = 18; fill in 18 – 7 = 11; |M  E| = 15; fill in 15 – 7 = 8. cont’d Figure 2.16b Second step is two-region intersections

27. 27 Example 5 – Solution Step 3 Fill in the other regions (see Figure 2.16c). | H | = 40, but 28 have previously been accounted for in the set H, so there are an additional 12 members (40 – 11 – 7 – 10 = 12). | E | = 41; fill in 41 – 8 – 7 – 10 = 16; | M | = 45; fill in 45 – 11 – 7 – 8 = 19. cont’d Third step is the one-region parts Figure 2.16c

28. 28 Example 5 – Solution Step 4 Add all the numbers listed in the sets of the Venn diagram to see that 83 students have been accounted for. Since 100 students were surveyed, we see that 17 are not taking any of the three courses. (We also filled this number in Figure 16c.) We now have the answers to the questions directly from the Venn diagram: a. 19 b. 16 c. 12 d. 17 Look Back. Does our answer make sense? Add all the numbers in the Venn diagram as a check to see that we have accounted for the 100 students. cont’d