1. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.2 Subsets

2. Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Subsets and proper subsets 2.2-2

3. 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

4. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

5. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

6. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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).

7. 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

8. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

9. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Number of Distinct Subsets The number of distinct subsets of a finite set A is 2 n, where n is the number of elements in set A. 2.2-9

10. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

11. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Solution: Since there are 4 elements in the given set, the number of distinct subsets is 2 4 = 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

12. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Number of Distinct Proper Subsets The number of distinct proper subsets of a finite set A is 2 n – 1, where n is the number of elements in set A. 2.2-12

13. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Number of Distinct Proper Subsets Example: Determine the number of distinct proper subsets for the given set {t, a, p, e}. 2.2-13

14. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Solution: The number of distinct proper subsets is 2 4 – 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