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Section 1.5

Section 1.5. Venn Diagrams and Set Operations. Objectives. Understand the meaning of a universal set and the basic ideas of a Venn diagram. Use Venn diagrams to visualize relationships between two sets. Find the complement of a set. Find the intersection and union of two sets.

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Section 1.5

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  1. Section 1.5 Venn Diagrams and Set Operations

  2. Objectives • Understand the meaning of a universal set and the basic ideas of a Venn diagram. • Use Venn diagrams to visualize relationships between two sets. • Find the complement of a set. • Find the intersection and union of two sets. • Perform operations with sets. • Determine sets involving set operations from a Venn diagram. • Understand the meaning of andandor. • Use the formula for n(A B).

  3. Key Terms • Universal Set: a set that contains all the elements for any specific discussion, symbolized by . • Venn Diagrams: (named for British logician, John Venn) a rectangle is drawn to represent the . • Complement of a Set:complement of Set A is the set of all elements in the set that are not in set A; symbolized by A’. • Intersection of Sets:intersection of set A and B is the set of elements that A and B have in common, symbolized by A∩B.

  4. Key Terms (Continued) • Union of Sets:the union of sets A and B, symbolized by AB, is the set of elements that are members of either A or B (or both). • And and or: the word “or” generally means union. The word “and” generally means intersection.

  5. Sets Take on Different Forms Disjoint B A A B Proper Equal Overlapping A A = B Sets with some common elements. B

  6. Overlapping Sets Four Regions A B Region I: elements in set A only.Region II: elements in set A and set B Region III: elements in set B only.Region IV: elements that do not belong in set A or set B.

  7. Note: • For any set A: • A ∩ = • A = A • Performing Set Operations:always begin by performing set operations inside parentheses; or just identify the elements in each set.

  8. Example 1: • Describe the Universal set that includes all elements in the given sets. • Set A= {Wm. Shakespeare, Charles Dickens}Set B = {Mark Twain, Robert Louis Stevenson} • Set A = {Pepsi, Sprite, Dr. Pepper}Set B = {Coca Cola, Seven-Up}

  9. Example 2: • U = {a, b, c, d, e, f, g}, A = {a, b, f, g}, B = {c, d, e}, C = {a, g}, and D = {a, b, c, d, e, f} • Find B’ • Find C’

  10. Example 3: A ∩ B • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6} • Find A • Find B • Find ∩

  11. Example 4: BC • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6} • Find B • Find C • Find

  12. Example 5: B’ ∩ C • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6} • Find B’ C • Find ∩

  13. Section 1.5 Assignments • Classwork: • TB pg. 46/1 - 10 • Must write problems and show ALL work to receive credit for this assignment. • Homework: • Create Engrade Account

  14. Example 6: A’ B’ • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6} • Find A’ • Find B’ • Find A’ B’

  15. Example 7: A’ (B ∩ C) • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6} • Find B • Find C • Find ∩ • Find A’ • Find

  16. Example 8: • In order to increase its readership, a computer magazine conducted a survey of people who have recently purchased a new computer and identified the following groups: • E = {x/x will use the computer for education}, B = {x/x will use the computer for business}, H = {y/y will use the computer for home management} • Use this information to describe verbally the following set. • E ∩ H

  17. Example 9: • Using the same information from Example 8. • (EH) ∩ B

  18. Key Terms • Difference of Sets: the set of elements that are in B but not in A. This is denoted by B – A.

  19. Example 10: Using Set Difference • Find {3, 6, 9, 12, 15} – {x/x is an odd integer} • M = {jo, st}, W = {ba, be, ca, st}…Find M – W

  20. Section 1.5 Assignments • Classwork: • TB pg. 46/13 – 24 all • Must write problems and show ALL work to receive credit for this assignment. • Homework: • Do not forget to create Engrade account.

  21. Section 1.5 con’t Venn Diagrams and Set Operations with Three Sets

  22. Three Sets – 8 Regions II V IV VI VIII

  23. DeMorgan’s Law • (A B)’ = A’ ∩ B’ • (A ∩ B)’ = A’ B’

  24. Example 11: • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}

  25. Example 12: • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}

  26. Example 13: • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}

  27. Example 14: • Which regions represent set C?

  28. Example 15: • Which regions represent B C?

  29. Example 16: • Use the Venn diagram to represent each set in roster form. 4 5 6 7 8 9

  30. Example 17: • Use the Venn diagram to represent the set in roster form. 4 5 6 7 8 9

  31. Example 18: • Construct a Venn diagram using the following information.

  32. Example 19: • Determine if the sets are equal using a Venn diagram.

  33. Example 20: • Determine if the sets are equal using a Venn diagram.

  34. Section 1.5 Assignments • Classwork: • TB pg. 47/26 – 44 Even, and 57 – 64 All • Must write problems and show ALL work to receive credit for this assignment. • Homework: • Do not forget to create Engrade account.

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