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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.
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
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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
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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
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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
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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
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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
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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
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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
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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
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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 '
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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
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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
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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
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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
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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
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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
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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
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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
