Section 2 2 Subsets Copyright 2013 2010 2007

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

Determining Subsets Example: Determine whether set A is a subset of set B. A = { 3, 5, 6, 8, 11 } B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Solution: All of the elements of set A are not contained in set B, so A ⊈ B. 2. 2 -5 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Try This: Determine if Set M is a subset of set N Set M = {1, 3 , 7} Set N = { 1, 2, 3, 6 , 7, 9} 2. 2 -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). 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 -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 -9 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Try This: is Set X a proper subset of Y? Set X = { A, B, C, D} Set Y = { A, B, C, …, Z} 2. 2 -10 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 -11 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 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Number of Distinct Subsets Solution: Since there are 4 elements in the given set, the number of distinct subsets is 24 = 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}, { } 2. 2 -13 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Try This: Determine the number of subsets and list them. {a, b, c} 2. 2 -14 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 -15 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 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Number of Distinct Subsets Solution: The number of distinct proper subsets is 24 – 1= 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. 2. 2 -17 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Try this: Determine the number of distinct proper subsets {a, b, c} 2. 2 -18 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Homework P. 55 -56 # 1 – 4 all, 6 – 57 (x 3) 2. 2 -19 Copyright 2013, 2010, 2007, Pearson, Education, Inc.