Section 2 3 Venn Diagrams and Set Operations

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Section 2. 3 Venn Diagrams and Set Operations Copyright 2013, 2010, 2007, Pearson, Education,

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,

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. •

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. • 2. 3 -3 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Venn Diagrams A rectangle usually represents the universal set, U. • The items inside

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. • 2. 3 -4 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Disjoint Sets Two sets which have no elements in common are said to be

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. • 2. 3 -5 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Proper Subset 2. 3 -6 • If set A is a proper subset of

Proper Subset 2. 3 -6 • 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

Equal Sets If set A contains exactly the same elements as set B, A = B. • Both sets are drawn as one circle. • 2. 3 -7 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Overlapping Sets 2. 3 -8 • Two sets A and B with some elements

Overlapping Sets 2. 3 -8 • 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 A and B, are disjoint, they have no elements in

Case 1: Disjoint Sets A and B, are disjoint, they have no elements in common. • Region II is empty. • 2. 3 -9 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Case 2: Subsets 2. 3 -10 • When A ⊆ B, every element of

Case 2: Subsets 2. 3 -10 • 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

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. • 2. 3 -11 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Case 4: Overlapping Sets When sets A and B have elements in common, those

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. • 2. 3 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Region IV • 2. 3 -13 In each of the four cases, any element

Region IV • 2. 3 -13 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 • 2. 3 -14 The complement of set A, symbolized

Complement of a Set • 2. 3 -14 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.

Example 1: A set and Its Complement Given U = {1, 2, 3, 4,

Example 1: A set and Its Complement 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. 2. 3 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 1: A set and Its Complement Solution U = {1, 2, 3, 4,

Example 1: A set and Its Complement 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}. 2. 3 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Intersection The intersection of sets A and B, symbolized A ∩ B, is the

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. • 2. 3 -17 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Intersection of Sets Given U = {1, 2, 3, 4, 5, 6,

Example 3: Intersection of Sets 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 ⋂ B c) A´⋂ B 2. 3 -18 b) A ⋂ C d) (A ⋂ B)´ Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Intersection of Sets Solution U = {1, 2, 3, 4, 5, 6,

Example 3: Intersection of Sets 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. 2. 3 -19 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Intersection of Sets Solution U = {1, 2, 3, 4, 5, 6,

Example 3: Intersection of Sets 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. 2. 3 -20 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Intersection of Sets Solution U = {1, 2, 3, 4, 5, 6,

Example 3: Intersection of Sets 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´ ⋂ B = {4, 5, 6, 7, 9, 10} ⋂ {1, 3, 6, 7, 8} = {6, 7} 2. 3 -21 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Intersection of Sets Solution U = {1, 2, 3, 4, 5, 6,

Example 3: Intersection of Sets 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 = {1, 3, 8} (A ⋂ B)´= {1, 3, 8}´ = {2, 4, 5, 6, 7, 9, 10} 2. 3 -22 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Try This: Use the information to find the solutions U = {a, b, c,

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} 2. 3 -23 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Union • The union of sets A and B, symbolized A ⋃ B, is

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. 2. 3 -24 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: The Union of Sets Given U = {1, 2, 3, 4, 5,

Example 5: The Union of Sets 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 ⋃ B c) A´⋃ B 2. 3 -25 b) A ⋃ C d) (A ⋃ B)´ Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: The Union of Sets Solution U = {1, 2, 3, 4, 5,

Example 5: The Union of Sets 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} 2. 3 -26 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: The Union of Sets Solution U = {1, 2, 3, 4, 5,

Example 5: The Union of Sets 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. 2. 3 -27 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: The Union of Sets Solution U = {1, 2, 3, 4, 5,

Example 5: The Union of Sets 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´ ⋃ B = {3, 5, 7, 8, 9, 10} ⋃ {1, 3, 6, 7, 9} = {1, 3, 5, 6, 7, 8, 9, 10} 2. 3 -28 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: The Union of Sets Solution U = {1, 2, 3, 4, 5,

Example 5: The Union of Sets 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 = {1, 2, 3, 4, 6, 7, 9} (A ⋃ B)´= {1, 2, 3, 4, 6, 7, 9}´ = {5, 8, 10} 2. 3 -29 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Try This: Use the information to find the solutions U = {a, b, c,

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} 2. 3 -30 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Homework p. 64 # 9 – 69 (x 3) Ch. 2. 1 – 2.

Homework p. 64 # 9 – 69 (x 3) Ch. 2. 1 – 2. 2 Quiz next class 2. 3 -31 Copyright 2013, 2010, 2007, Pearson, Education, Inc.