Section 2.3 Venn Diagrams and Set Operations

1. Section 2.3Venn Diagrams and Set Operations

2. What You Will Learn • Venn diagrams

3. Venn Diagrams • A Venn diagram is a useful technique for illustrating set relationships. • Named for John Venn. • Venn invented and used them to illustrate ideas in his text on symbolic logic.

4. Venn Diagrams • A rectangle usually represents the universal set, U. • The items inside the rectangle may be divided into subsets of U and are represented by circles. • The circle labeled A represents set A.

5. Disjoint Sets • Two sets which have no elements in common are said to be disjoint. • The intersection of disjoint sets is the empty set. • There are no elements in common since there is no overlapping area between the two circles.

6. Proper Subset • If set A is a proper subset of set B, A ⊂ B. • Circle A is completely inside circle B.

7. Equal Sets • If set A contains exactly the same elements as set B, A = B. • Both sets are drawnas one circle.

8. Overlapping Sets • Two sets A and B with some elements in common. • This is the most general form of a Venn Diagram.

9. Case 1: Disjoint Sets • Sets A and B, are disjoint, they have no elements in common. • Region II is empty.

10. Case 2: Subsets • When A ⊆ B, every element of set A is also an element of set B. • Region I is empty. • If B ⊆ A, however, then region III is empty.

11. Case 3: Equal Sets • When set A = set B, all elements of set A are elements of set B and all • elements of set B are elements of set A. • Regions I and III are empty.

12. Case 4: Overlapping Sets • When sets A and B have elements in common, those elements are in region II. • Elements that belong to set A but not to set B are in region I. • Elements that belong to set B but not to set A are in region III.

13. Region IV • In each of the four cases, any element belonging to the universal set but not belonging to set A or set B is placed in region IV.

14. Complement of a Set • The complement of set A, symbolized A´, is the set of all elements in theuniversal set thatare not in set A.

15. Example 1: A set and Its Complement Given U = {1, 2, 3, 4, 5, 6, 7, 8} andA = { 1, 3, 4} Find A and illustrate the relationship among sets U, A, and A´ in a Venn diagram.

16. Example 1: A set and Its Complement • Solution U = {1, 2, 3, 4, 5, 6, 7, 8} andA = { 1, 3, 4} All of the elements in U that are not in set A are 2, 5, 6, 7, 8. Thus,A´ = {2, 5, 6, 7, 8}.

17. Intersection • The intersection of sets A and B, symbolized A ∩ B, is the set containing all the elements that • are common to both set A and set B. • Region II represents the intersection.

18. Example 3: Intersection of Sets Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = { 1, 2, 3, 8} B = {1, 3, 6, 7, 8} C = { } Find a) A⋂ B b) A⋂C c) A´⋂B d) (A⋂B)´

19. Example 3: Intersection of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = { 1, 2, 3, 8} B = {1, 3, 6, 7, 8} C = { } a) A⋂B = {1, 2, 3, 8}⋂ {1, 3, 6, 7, 8} The elements common to both set A and B are 1, 3, and 8.

20. Example 3: Intersection of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = { 1, 2, 3, 8} B = {1, 3, 6, 7, 8} C = { } b) A ⋂ C = {1, 2, 3, 8}⋂ { } There are no elements common to both set A and C.

21. Example 3: Intersection of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = { 1, 2, 3, 8} B = {1, 3, 6, 7, 8} c) A´⋂ B First determine A´ A´= {4, 5, 6, 7, 9,10} A´⋂ B = {4, 5, 6, 7, 9,10} ⋂{1, 3, 6, 7, 8} = {6, 7}

22. Example 3: Intersection of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = { 1, 2, 3, 8} B = {1, 3, 6, 7, 8} d) (A ⋂ B)´First determine A ⋂B A⋂B = {1, 3, 8} (A ⋂ B)´= {1, 3, 8}´ = {2, 4, 5, 6, 7, 9, 10}

23. Try This: Use the information to find the solutions U = {a, b, c, d, e, f, g, h} A = { a, d, h} B = {b, c, d, e}

24. Union • The union of sets A and B, symbolized A ⋃B, is the set containing all the elements that • are members ofset A or of set B (or of both sets). • Regions I, II, and III represents the union.

25. Example 5: The Union of Sets Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = { 1, 2, 4, 6} B = {1, 3, 6, 7, 9} C = { } Find a) A⋃B b) A⋃C c) A´⋃B d) (A⋃B)´

26. Example 5: The Union of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = {1, 2, 4, 6} B = {1, 3, 6, 7, 9} C = { } a) A⋃B = {1, 2, 4, 6}⋃{1, 3, 6, 7, 9} = {1, 2, 3, 4, 6, 7, 9}

27. Example 5: The Union of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = {1, 2, 4, 6} B = {1, 3, 6, 7, 9} C = { } b) A⋃C = {1, 2, 4, 6}⋃ { } = {1, 2, 4, 6} Note that A⋃C = A.

28. Example 5: The Union of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = {1, 2, 4, 6} B = {1, 3, 6, 7, 9} c) A´⋃B First determine A´ A´= {3, 5, 7, 8, 9, 10} A´⋃B = {3, 5, 7, 8, 9, 10}⋃ {1, 3, 6, 7, 9} = {1, 3, 5, 6, 7, 8, 9, 10}

29. Example 5: The Union of Sets • Solution U = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10}A = {1, 2, 4, 6} B = {1, 3, 6, 7, 9} d) (A⋃ B)´First determine A⋃B A⋃B = {1, 2, 3, 4, 6, 7, 9} (A⋃B)´= {1, 2, 3, 4, 6, 7, 9}´ = {5, 8, 10}

30. Try This: Use the information to find the solutions U = {a, b, c, d, e, f, g, h} A = { a, d, h} B = {b, c, d, e}

31. Homework p. 64 # 9 – 69 (x 3) Ch. 2.1 – 2.2 Quiz next class