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2.4

2.4. Venn Diagrams with Three Sets. When you have elements to place in three sets, work from the inside out Step 1: Find the elements that are common to all three sets and place in region V. General Procedure for Constructing Venn Diagrams with Three Sets. Step 2:

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2.4

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  1. 2.4 Venn Diagrams with Three Sets

  2. When you have elements to place in three sets, work from the inside out Step 1: Find the elements that are common to all three sets and place in region V. General Procedure for Constructing Venn Diagrams with Three Sets

  3. Step 2: Find the elements for region II. Find the elements in . The elements in this set belong in regions II and V. Place the elements in the set that are not listed in region V in region II. The elements in regions IV and VI are found in a similar manner. General Procedure for Constructing Venn Diagrams with Three Sets continued

  4. Step 3: Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V. The elements in regions III and VII are found in a similar manner. General Procedure for Constructing Venn Diagrams with Three Sets continued

  5. Step 4: Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII. TIP: As you work the examples and problems, print out Steps 1-4 from the slides. General Procedure for Constructing Venn Diagrams with Three Sets continued U II I III A V B IV VI VII VIII C

  6. Example: Constructing a Venn diagram for Three Sets Construct a Venn diagram illustrating the following sets. U = {1, 2, 3, 4, 5, 6, 7, 8} A = { 1, 2, 5, 8} B = {2, 4, 5} C = {1, 3, 5, 8} Solution: • Find the intersection of all three sets and place in region V, {5}. Click on icon to Play Audio

  7. Example: Constructing a Venn diagram for Three Sets continued • Determine the intersection of sets A and B and place in region II. {2, 5} Element 5 has already been placed in region V, so 2 must be placed in region II. • Now determine the numbers that go into region V. { 1, 2, 5, 8} Since 5 has been placed in region V, place 1 and 8 in region IV.

  8. Example: Constructing a Venn diagram for Three Sets continued • Now determine the numbers that go in region VI. {5} • There are no new numbers to be placed in this region. Since all numbers in set A have been placed, there are no numbers in region I. The same procedures using set B completes region III. Using set C completes region VII.

  9. U 4 2 II I III 5 A 1,8 V B IV VI 3 VII VIII C 6 7 Example: Constructing a Venn diagram for Three Sets continued • The Venn diagram is then completed.

  10. De Morgan’s Laws • Skip this section

  11. Next Steps • Read Examples 1-2 • Work Problems in text on p. 71: 9--12, all; 23-34, all • Do Online homework corresponding to this section

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