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Set Operations

Set Operations. MATH 102 Contemporary Math S. Rook. Overview. Section 2.3 in the textbook: Intersection & Union Complement Difference. Intersection & Union. Intersection of Sets.

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Set Operations

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  1. Set Operations MATH 102 Contemporary Math S. Rook

  2. Overview • Section 2.3 in the textbook: • Intersection & Union • Complement • Difference

  3. Intersection & Union

  4. Intersection of Sets • Given sets A and B, the intersection of A and B, denoted , means to list those elements common to both sets • Only those elements present in BOTHA and B are part of the intersection • e.g. Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. Find • Can also represent using a Venn Diagram

  5. Union of Sets • Given sets A and B, the union of A and B, denoted , means to combine the elements of A and B together • i.e. fill an initially empty set (bag) with the elements of A and then the elements of B • An element present in BOTHA and B is only added once to the union • e.g. Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. Find A U B • Can also represent using a Venn Diagram

  6. Union of Sets (Continued) • Given sets A and B, what is the relationship between n(A), n(B), and n(A U B)? • e.g. Let A = {p| p is a person sitting in the front row} and B = {g | g is a person wearing glasses}. What are the values of n(A) and n(B)? • What is the value of n(A U B)? • Why does the sum of n(A) and n(B) not equal n(A U B)? • Verify for yourself that the value of n(A U B) checks for sets A and B on the previous slide

  7. Intersection & Union (Example) Ex 1: Let A = {2, 3, 4, 5, 7, 9}, B = {x | x is an even natural number}, and C = {y | y is an odd natural number} . Find the following sets: a) b) c)

  8. Complement

  9. Complement • Universal set: the set of all elements being considered in a problem. Often denoted by U. • All subsets in a problem are taken from the universal set • Given set A, the complement of A, denoted by A’, means to list those elements that A is missing from the universal set • i.e. those elements that need to be added to A to complete the universal set • e.g. Let U = {1, 2, 3, … , 10} and A = {1, 6, 9, 10}. Find A’. • Can also represent using a Venn Diagram

  10. Complement (Example) Ex 2: Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, g}, B = {a, b, c, d, e, f, g, h}, and C = { }. Find: a) A’ b) B’ c) C’

  11. Difference

  12. Difference • Given sets A and B, the difference of A and B, denoted A – B, means the resulting set when the elements of B are removed from the elements of A • i.e. Just like subtraction, we are taking away those elements in B away from A • e.g. Let A = {1, 2, 3, 4, 5, 6} and B = {2, 3, 4}. Find A – B. • Can also represent using a Venn Diagram

  13. Difference (Example) Ex 3: Let U = {1, 2, 3, … }, A = {1, 3, 5, 7, 9, … }, B = {1, 2, 3, 4, 5, 6, 8}, and C = {2, 4, 6, 8}. Find the resulting set: a) B – C b) C – B c) A – C

  14. Combining Set Operations (Example) Ex 4: Let U = {1, 2, 3, …, 10 }, A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5, 6}, and C = {2, 4, 6, 7, 8}. Find the resulting set: a) b)

  15. Combining Set Operations (Example) Ex 5: Shade the appropriate regions in a Venn Diagram to represent the resulting set: a) b)

  16. Summary • After studying these slides, you should know how to do the following: • Find the intersection and union of sets • Calculate the number of elements in the union of sets • Find the complement and difference of sets • Apply multiple set operations • Use Venn Diagrams to illustrate set operations • Additional Practice: • See the list of suggested problems for 2.3 • Next Lesson: • Survey Problems (Section 2.4)

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