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Section 2.2 Subsets

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.

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Section 2.2 Subsets

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  1. Section 2.2Subsets

  2. What You Will Learn • Subsets and proper subsets

  3. 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.”

  4. 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.

  5. Determining Subsets 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.

  6. 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. 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.

  8. 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.

  9. 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.

  10. 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}.

  11. Number of Distinct Subsets 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}, { }

  12. Number of Distinct Proper Subsets • The number of distinct proper subsets of a finite set A is 2n– 1, where n is the number of elements in set A.

  13. Number of Distinct Proper Subsets Example: Determine the number of distinct proper subsets for the given set{t, a, p, e}.

  14. Number of Distinct Subsets 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.

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