Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.3 Venn Diagrams and Set Operations

Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Venn diagrams 2.3-2

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Proper Subset If set A is a proper subset of set B, A ⊂ B. Circle A is completely inside circle B

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Equal Sets If set A contains exactly the same elements as set B, A = B. Both sets are drawn as one circle

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Overlapping Sets Two sets A and B with some elements in common. This is the most general form of a Venn Diagram

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Case 1: Disjoint Sets Sets A and B, are disjoint, they have no elements in common. Region II is empty

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Complement of a Set The complement of set A, symbolized A ´, is the set of all elements in the universal set that are not in set A

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Given U = {1, 2, 3, 4, 5, 6, 7, 8} and A = { 1, 3, 4} Find A and illustrate the relationship among sets U, A, and A ´ in a Venn diagram. Example 1: A set and Its Complement

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Solution U = {1, 2, 3, 4, 5, 6, 7, 8} and A = { 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}. Example 1: A set and Its Complement

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 ⋂ Bb) A ⋂ C c) A ´ ⋂ Bd) (A ⋂ B) ´ Example 3: Intersection of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. Example 3: Intersection of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. Example 3: Intersection of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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´ ⋂ BA´ ⋂ B = {4, 5, 6, 7, 9,10} ⋂ {1, 3, 6, 7, 8} = {6, 7} Example 3: Intersection of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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} Example 3: Intersection of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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}

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Union The union of sets A and B, symbolized A ⋃ B, is the set containing all the elements that are members of set A or of set B (or of both sets). Regions I, II, and III represents the union

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 ⋃ Bb) A ⋃ C c) A ´ ⋃ Bd) (A ⋃ B) ´ Example 5: The Union of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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} Example 5: The Union of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. Example 5: The Union of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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´ ⋃ BA´ ⋃ B = {3, 5, 7, 8, 9, 10} ⋃ {1, 3, 6, 7, 9} = {1, 3, 5, 6, 7, 8, 9, 10} Example 5: The Union of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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} Example 5: The Union of Sets

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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}

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Homework p. 64 # 9 – 69 (x 3) Ch. 2.1 – 2.2 Quiz next class