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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.3B Venn Diagrams and Set Operations
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 2.3B
Venn Diagrams and Set
Operations
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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 }
2.3-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.3-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
The Number of Elements in
A ⋃ B
For any finite sets A and B,
n(A ⋃ B) = n(A) + n(B) – n(A ∩ B)
2.3-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Try This: Find n(A U B)
U = {a, b, c, d, e, f, g, h}A = { a, d, h}B = {b, c, d, e}
2.3-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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?
2.3-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionSet A is visitors who speak SpanishSet B is visitors who speak FrenchWe need to determine A ⋃ B.
n(A ⋃ B) = n(A) + n(B) – n(A ∩ B)n(A ⋃ B) = 25 + 14 – 4
= 35
Example 7: How Many Visitors Speak Spanish or French?
2.3-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.3-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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 }
2.3-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
GivenU = {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}
Finda) A – B b) A – Cc) A´– B d) A – C´
Example 9: The Difference of Two Sets
2.3-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionU = {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
2.3-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionU = {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
2.3-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionU = {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
2.3-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionU = {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}
Example 9: The Difference of Two Sets
2.3-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Try This: Use the information to find the solutionsU = {a, b, c, d, e, f, g, h}A = { a, d, h}B = {b, c, d, e}
2.3-15
BA BA '
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.3-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.3-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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 10: The Cartesian Product of Two Sets
2.3-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionA = {orange, banana, apple} andB = {1, 2}
a) A B = {(orange, 1), (orange, 2), (banana, 1), (banana, 2), (apple, 1), (apple, 2)}
Example 5: The Union of Sets
2.3-19
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionA = {orange, banana, apple} andB = {1, 2}
b) B A = {(1, orange), (1, banana), (1, apple), (2, orange), (2, banana), (2, apple)}
Example 5: The Union of Sets
2.3-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionA = {orange, banana, apple} andB = {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
2.3-21
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionA = {orange, banana, apple} andB = {1, 2}
d) B B = {(1, 1), (1, 2), (2, 1), (2, 2)}
Example 5: The Union of Sets
2.3-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Try This: Use the information to find the solutionsU = {a, b, c, d, e, f, g, h}A = { a, d, h}B = {b, c, d, e}
2.3-23
BA AB
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Homework
p. 68 # 71 – 84 all, 85 – 110 (x5)
2.3-24
