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Discrete Mathematics

Discrete Mathematics. Section 5.03-Venn Diagrams and Set Operations ST 2. Objective: The students will…. • use Venn diagrams to represent sets • use set operations. Venn Diagrams . A Venn diagram is a technique used for picturing set relationships.

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Discrete Mathematics

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  1. Discrete Mathematics Section 5.03-Venn Diagrams and Set Operations ST 2

  2. Objective: The students will… • use Venn diagrams to represent sets • use set operations

  3. Venn Diagrams • A Venn diagram is a technique used for picturing set relationships. • A rectangle usually represents the universal set, U. • The items inside the rectangle are divided into subsets and are represented by circles.

  4. Four ways to represent sets in a Venn Diagram… • Disjoint Sets • Proper Subsets • Equal Sets • Overlapping Sets • Note: In each of the four cases, any element not belonging to set A or set B is placed in the universal set U

  5. Disjoint Sets • Two sets which have no elements in common are said to be disjoint.

  6. U A B Proper Subsets • Everything in set B is contained in set A so B is a proper subset of A,

  7. U A B Equal Sets • Both sets contain exactly the same elements, A = B

  8. Set A and Set B have some elements in common but each has extra elements in it that the other doesn’t. Overlapping Sets

  9. Complement of a Set • The set known as the complement contains all the elements of the universal set which are not part of the subsets. • Notation: A´ (A prime) is used to represent the complement set.

  10. U A Example • Let U = {2, 4, 6, 8, 10, 12, 14} and let A = {2, 6,10}. What is the complement of A? • Answer: A’ = {4, 8, 12, 14} (all the things which are not in set A but are part of the universal set) 2 6 10 4 8 12 14

  11. Intersection (aka “and”) • The intersection of two sets contains only those elements common to both of the sets. • Notation: This symbol represents an intersection

  12. Example • Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, what is the intersection of A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8}? • Answer: A B = {4, 5, 6} The overlap region contains the solution to an intersection 1 2 3 4 5 6 7 8 9

  13. Union (aka “or”) • The union of two given sets contains all of the elements for those sets. • The union “unites” that is, it brings together everything into one set. • Notation: This symbol represents a union

  14. Example • Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, what is the union of A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8}? • Answer: A B = {1, 2, 3, 4, 5, 6, 7, 8} The solution comes from merging the sets together ( we do not have to repeat numbers) 1 2 3 4 5 6 7 8 9

  15. If U= {a, b, c, d, e, f, g, h}, A= {a, b, c}, B = {b, d, e, f, g}, find the following? Answer: 1. {a, b, c, d, e, f, g} 2. {b} 3. {a, c, d, e, f, g, h} You Try • A B • A B • (A B)’ U A B d, e, f, g b a, c h

  16. 5.03 Worksheet Assignment

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