section 2.2 subsets
DESCRIPTION
Section 2.2 Subsets. What You Will Learn. Subsets and proper subsets. Subsets. Set A is a subset of set B , symbolized A ⊆ B , if and only if all elements of set A are also elements of set B . The symbol A ⊆ B indicates that “set A is a subset of set B .”. Subsets. - PowerPoint PPT PresentationTRANSCRIPT
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Section 2.2
Subsets
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Subsets and proper subsets
2.2-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Subsets
• Set A is a subset of set B, symbolized A ⊆ B, if and only if all elements of set A are also elements of set B.
• The symbol A ⊆ B indicates that “set A is a subset of set B.”
2.2-3
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Subsets
• The symbol A ⊈ B set A is not a subset of set B.
• To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B.
2.2-4
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Example: Determine whether set A is a subset of set B.A = { 3, 5, 6, 8 }B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Solution: All of the elements of set A are contained in set B, so A ⊆ B.
Determining Subsets
2.2-5
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Proper Subset
Set A is a proper subset of set B, symbolized A ⊂ B, if and only if all of the elements of set A are elements of set B and set A ≠ B (that is, set B must contain at least one element not is set A).
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Determining Proper Subsets
Example: Determine whether set A is a proper subset of set B.A = { dog, cat }B = { dog, cat, bird, fish }
Solution: All the elements of set A are contained in set B, and sets A and B are not equal, therefore A ⊂ B.
2.2-7
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Determining Proper Subsets
Example: Determine whether set A is a proper subset of set B.A = { dog, bird, fish, cat }B = { dog, cat, bird, fish }
Solution: All the elements of set A are contained in set B, but sets A and B are equal, therefore A ⊄ B.
2.2-8
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Number of Distinct Subsets
The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A.
2.2-9
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Number of Distinct Subsets
Example:
Determine the number of distinct subsets for the given set {t, a, p, e}.
List all the distinct subsets for the given set {t, a, p, e}.
2.2-10
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Solution: Since there are 4 elements in the
given set, the number of distinct subsets is
24 = 2 • 2 • 2 • 2 = 16.{t,a,p,e}, {t,a,p}, {t,a,e}, {t,p,e},
{a,p,e},{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { }
Number of Distinct Subsets
2.2-11
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Number of Distinct Proper SubsetsThe number of distinct proper subsets of a finite set A is 2n – 1, where n is the number of elements in set A.
2.2-12
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Number of Distinct Proper SubsetsExample:
Determine the number of distinct proper subsets for the given set{t, a, p, e}.
2.2-13
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Solution: The number of distinct proper subsets is 24 – 1= 2 • 2 • 2 • 2 – 1 = 15.They are {t,a,p}, {t,a,e}, {t,p,e},
{a,p,e},{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { }.
Only {t,a,p,e}, is not a proper subset.
Number of Distinct Subsets
2.2-14