Section 2.3 Venn Diagrams and Set Operations

What You Will Learn Venn diagrams

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.

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.

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.

Proper Subset If set A is a proper subset of set B, A ⊂ B. Circle A is completely inside circle B.

Equal Sets If set A contains exactly the same elements as set B, A = B. Both sets are drawn as one circle.

Overlapping Sets Two sets A and B with some elements in common. This is the most general form of a Venn Diagram.

Case 1: Disjoint Sets Sets A and B, are disjoint, they have no elements in common. Region II is empty.

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.

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.

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.

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.

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.

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.

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}.

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.

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 b) A ⋂ C c) A ⋂ B d) (A ⋂ B)

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.

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.

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}

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 A ⋂ B = {1, 3, 8} (A ⋂ B) = {1, 3, 8} = {2, 4, 5, 6, 7, 9, 10}

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.

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 b) A ⋃ C c) A ⋃ B d) (A ⋃ B)

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}

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.

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}

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 A ⋃ B = {1, 2, 3, 4, 6, 7, 9} (A ⋃ B) = {1, 2, 3, 4, 6, 7, 9} = {5, 8, 10}

The Meaning of and and or and is generally interpreted to mean intersection A ∩ B = { x | x  A and x  B } or is generally interpreted to mean union A ⋃ B = { x | x  A or x  B }

The Relationship Between n(A ⋃ B), n(A), n(B), n(A ∩ B) To find the number of elements in the union of two sets A and B, we add the number of elements in set A and B and then subtract the number of elements common to both sets.

The Number of Elements in A ⋃ B For any finite sets A and B, n(A ⋃ B) = n(A) + n(B) – n(A ∩ B)

Example 7: How Many Visitors Speak Spanish or French? The results of a survey of visitors at the Grand Canyon showed that 25 speak Spanish, 14 speak French, and 4 speak both Spanish and French. How many speak Spanish or French?

Example 7: How Many Visitors Speak Spanish or French? Solution Set A is visitors who speak Spanish Set B is visitors who speak French We need to determine A ⋃ B. n(A ⋃ B) = n(A) + n(B) – n(A ∩ B) n(A ⋃ B) = 25 + 14 – 4 = 35

Difference of Two Sets The difference of two sets A and B, symbolized A – B, is the set of elements that belong to set A but not to set B. Region 1 represents the difference of the two sets.

Difference of Two Sets Using set-builder notation, the difference between two sets A and B is indicated by A – B = { x | x  A or x  B }

Example 9: The Difference of Two Sets Given U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h} B = {a, b, d, h, i} C = {b, e, g} Find a) A – B b) A – C c) A– B d) A – C

Example 9: The Difference of Two Sets Solution U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h} B = {a, b, d, h, i} C = {b, e, g} a) A – B is the set of elements that are in set A but not in set B. A – B = {e, f, g}

Example 9: The Difference of Two Sets Solution U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h} B = {a, b, d, h, i} C = {b, e, g} b) A – C is the set of elements that are in set A but not in set C. A – C = {d, f, h}

Example 9: The Difference of Two Sets Solution U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h} B = {a, b, d, h, i} c) A– B is the set of elements that are in set A but not in set B. A = {a, c, i, j, k} A – B = {c, j, k}

Example 9: The Difference of Two Sets Solution U = {a, b, c, d, e, f, g, h, i, j, k} A = {b, d, e, f, g, h} C = {b, e, g} c) A – C is the set of elements that are in set A but not in set C. C = {a, c, d, f, h, i, j, k} A – C = {b, e, g}

Cartesian Product The Cartesian product of set A and set B, symbolized A × B, and read “A cross B,” is the set of all possible ordered pairs of the form (a, b), where a  A and b  B.

Ordered Pairs in a Cartesian Product Select the first element of set A and form an ordered pair with each element of set B. Then select the second element of set A and form an ordered pair with each element of set B. Continue in this manner until you have used each element in set A.

Example 10: The Cartesian Product of Two Sets Given A = {orange, banana, apple} and B = {1, 2}, determine the following. a) A × B b) B × A c) A × A d) B × B

Example 5: The Union of Sets Solution A = {orange, banana, apple} and B = {1, 2} a) A × B = {(orange, 1), (orange, 2), (banana, 1), (banana, 2), (apple, 1), (apple, 2)}

Example 5: The Union of Sets Solution A = {orange, banana, apple} and B = {1, 2} b) B × A = {(1, orange), (1, banana), (1, apple), (2, orange), (2, banana), (2, apple)}

Example 5: The Union of Sets Solution A = {orange, banana, apple} and B = {1, 2} c) A × A = {(orange, orange), (orange, banana), (orange, apple), (banana, orange), (banana, banana), (banana, apple), (apple, orange), (apple, banana), (apple, apple)}

Example 5: The Union of Sets Solution A = {orange, banana, apple} and B = {1, 2} d) B × B = {(1, 1), (1, 2), (2, 1), (2, 2)}