Section 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets

1. Section 2.4Venn Diagrams with Three Sets and Verification of Equality of Sets

2. What You Will Learn • Venn Diagram with Three Sets • Verification of Equality of Sets

3. Three Sets: Eight Regions • When three sets overlap, it creates eight regions.

4. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • Determine the elements to be placed in region V by finding the elements that are common to all three sets, A ∩ B ∩ C.

5. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • Determine the elements to be placed in region II. Find the elements in A ∩ B and place the elements that are not listed in region V in region II.

6. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • Determine the elements to be placed in region IV. Find the elements in A ∩ C and place the elements that are not listed in region V in region IV.

7. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • Determine the elements to be placed in region VI. Find the elements in B ∩ C and place the elements that are not listed in region V in region VI.

8. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • 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.

9. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • Determine the elements to be placed in region III by determining the elements in set B that are not in regions II, V, and VI.

10. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • Determine the elements to be placed in region VII by determining the elements in set C that are not in regions IV, V, and VI.

11. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C • 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.

12. Example 2: Blood Types Human blood is classified (typed) according to the presence or absence of the specific antigens A, B, and Rh in the red blood cells. Antigens are highly specified proteins and carbohydrates that will trigger the production of antibodies in the blood to fight infection. Blood containing the Rh antigen is labeled positive, +, while blood lacking the Rh antigen is labeled negative, –.

13. Example 2: Blood Types Blood lacking both A and B antigens is called type O. Sketch a Venn diagram with three sets A, B, and Rh and place each type of blood listed in the proper region. A person has only one type of blood.

14. Example 2: Blood Types

15. Example 2: Blood Types • Solution Blood containing Rh is is + Blood not containing Rh is – All blood in the Rh circle is + All blood outside the Rh circle is – Intersection of all 3 sets, V, is AB+ II contains only A and B, AB–

16. Example 2: Blood Types • Solution I contains A only, A– III contains B only, B– IV is A+ VI is B+ VII contains only Rh antigen, O+ VIII lacks all three antigens, O–

17. Verification of Equality of Sets • To verify set statements are equal for any two sets selected, we use deductive reasoning with Venn Diagrams. • If both statements represent the same regions of the Venn Diagram, then the statements are true for all sets A and B.

18. Example 3: Equality of Sets Determine whether (A⋃B)´=A´⋂B´for all sets A and B.

19. Example 3: Equality of Sets Solution Draw a Venn diagram with two sets A and B. Label the regions as indicated.

20. Example 3: Equality of Sets Solution

21. Example 3: Equality of Sets Solution Both statements are represented by the same region, IV. Thus (A⋃B)´=A´⋂B´for all sets A and B.

22. De Morgan’s Laws • A pair of related theorems known as De Morgan’s laws make it possible to change statements and formulas into more convenient forms. • (A⋂B)´=A´⋃B´ • (A⋃B)´=A´⋂B´