VENN DIAGRAM.ppt
- 2. 1/15/2024 Section 2.3 2 Venn diagrams and Set Operations Objectives 1. Understand the meaning of a universal set. 2. Understand the basic ideas of a Venn diagram. 3. Use Venn diagrams to visualize relationships between two sets. 4. Find the complement of a set 5. Find the intersection of two sets. 6. Find the union of two sets. 7. Perform operations with sets. 8. Determine sets involving set operations from a Venn diagram. 9. Understand the meaning of and and or. 10. Use the formula for n (A U B).
- 3. 1/15/2024 Section 2.3 3 Universal Sets and Venn Diagrams • The universal set is a general set that contains all elements under discussion. • John Venn (1843 – 1923) created Venn diagrams to show the visual relationship among sets. • Universal set is represented by a rectangle • Subsets within the universal set are depicted by circles, or sometimes ovals or other shapes.
- 4. 1/15/2024 Section 2.3 4 Example 1 Determining Sets From a Venn Diagram • Use the Venn diagram to determine each of the following sets: a. U U = {□, ∆ , $, M, 5 } b. A A = {□ , ∆ } c. The set of elements in U that are not in A. {$, M, 5 }
- 5. 1/15/2024 Section 2.3 5 Representing Two Sets in a Venn Diagram Disjoint Sets: Two sets that have Equal Sets: If A = B then AB no elements in common. and B A. Proper Subsets: All elements of Sets with Some Common Elements set A are elements of set B. Some means “at least one”. The representing the sets must overlap.
- 6. 1/15/2024 Section 2.3 6 Example 2 Determining sets from a Venn Diagram Solutions: a. U = { a, b, c, d, e, f, g } b. B = {d, e } c. {a, b, c } d. {a, b, c, f, g } e. {d} • Use the Venn Diagram to determine: a. U b. B c. The set of elements in A but not B d. The set of elements in U that are not in B e. The set of elements in both A and B.
- 7. 1/15/2024 Section 2.3 7 The Complement of a Set
- 8. 1/15/2024 Section 2.3 8 Example 3 Finding a Set’s Complement • Let U = { 1, 2, 3, 4, 5, 5, 6, 8, 9} and A = {1, 3, 4, 7 }. Find A’. • Solution: Set A’ contains all the elements of set U that are not in set A. Because set A contains the elements 1,3,4,and 7, these elements cannot be members of set A’: A’ = {2, 5, 6, 8, 9}
- 9. 1/15/2024 Section 2.3 9 The Intersection and Union of Sets • The intersection of sets A and B, written A∩B, is the set of elements common to both set A and set B. This definition can be expressed in set-builder notation as follows: A∩B = { x | x A and xB} • The union of sets A and B, written AUB is the set of elements are in A or B or in both sets. This definition can be expressed in set-builder notation as follows: AUB = { x | x A or xB} • For any set A: – A∩Ø = Ø – AUØ = A
- 10. 1/15/2024 Section 2.3 10 Example 4 Finding the Intersection of Two Sets Find each of the following intersections: a.{7, 8, 9, 10, 11} ∩ {6, 8, 10, 12} {8, 10} a.{1, 3, 5, 7, 9} ∩ {2, 4, 6, 8} Ø a.{1, 3, 5, 7, 9} ∩ Ø Ø
- 11. 1/15/2024 Section 2.3 11 Example 5 Finding the Union of Sets Find each of the following unions: a. {7, 8, 9, 10, 11} U {6, 8, 10, 12} b. {1, 3, 5, 7, 9} U {2, 4, 6, 8} c. {1, 3, 5, 7, 9} U Ø a. {6, 7, 8, 9, 10, 11, 12} b. {1, 2, 3, 4, 5, 6, 7, 8, 9} c. {1, 3, 5, 7, 9}
- 12. 1/15/2024 Section 2.3 12 Example 6 Performing Set Operations a. (A U B)’ • Solution: A U B = {1, 3, 7, 8, 9, 10} (A U B)’ = {2, 4, 5, 6} b. A’ ∩ B’ • Solution A’ = {2, 4, 5, 6, 8, 10} B’ = {1, 2, 4, 5, 6, 9} A’ ∩ B’ = {2, 4, 5, 6 } Always perform any operations inside parenthesis first! Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } B = { 3, 7, 8, 10 } • Find
- 13. 1/15/2024 Section 2.3 13 Example 7 Determining Sets from a Venn Diagram Set to Determine Description of Set Regions in Venn Diagram a. A B set of elements in A or B or Both I,II,III b. (A B)’ set of elements in U that are not in A B IV c. A B set of elements in both A and B II d. (A B)’ set of elements in U that are not in A B I, III, IV e. A’ B set of elements that are not in A and are in B III f. A B’ set of elements that are in A or not in B or both I,II, IV
- 14. 1/15/2024 Section 2.3 14
- 15. 1/15/2024 Section 2.3 15 Sets and Precise Use of Everyday English • Set operations and Venn diagrams provide precise ways of organizing, classifying, and describing the vast array of sets and subsets we encounter every day. • Or refers to the union of sets • And refers to the intersection of sets
- 16. 1/15/2024 Section 2.3 16 Example 8 The Cardinal Number of the Union of Two Finite Sets • Some of the results of the campus blood drive survey indicated that 490 students were willing to donate blood, 340 students were willing to help serve a free breakfast to blood donors, and 120 students were willing to do both. How many students were willing to donate blood or serve breakfast?
- 17. 1/15/2024 Section 2.3 17 Example 8