2.2 Subsets 55 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. Number of Distinct Proper Subsets The number of distinct proper subsets of a finite set A is − 2 1, n where n is the number of elements in set A. Every set is a subset of itself, but no set is a proper subset of itself. Thus, the number of proper subsets will always be one less than the number of subsets that can be made from any given set. We summarize this concept in the following expression. Example 4 Distinct Subsets a) Determine the number of distinct subsets for the set {S, L, E, D}. b) List all the distinct subsets for the set {S, L, E, D}. c) How many of the distinct subsets are proper subsets? Solution a) Since the number of elements in the set is 4, the number of distinct subsets is = × × × = 2 2 2 2 2 16. 4 b) {S, L, E, D} {S, L, E} {S, L} {S} {} {S, L, D} {S, E} {L} {S, E, D} {S, D} {E} {L, E, D} {L, E} {D} {L, D} {E, D} c) There are 15 proper subsets. Every subset except {S, L, E, D} is a proper subset. 7 Now try Exercise 37 Example 5 Car Options Janelle is ordering a new car. She can order some, all, or none of the following options: leather interior, moon roof, navigation system, power windows, power seats, alarm system, and premium sound system. How many different variations of the set of options are possible? Solution Janelle can order the car with no options, any one option, any two options, any three options, and so on, up to seven options. One technique used in problem solving is to consider similar problems that you have solved previously. If you think about this problem, you will realize it is the same as asking how many distinct subsets can be made from a set with seven elements. The number of different variations of the set of options is the same as the number of possible subsets of a set with seven elements. There are 2 ,7 or 128, possible subsets of a set with seven elements. Thus, there are 128 possible variations of the set of options for the car. 7 Now try Exercise 55 Instructor Resources for Section 2.2 in MyLab Math • Objective-Level Videos 2.2 • PowerPoint Lecture Slides 2.2 • MyLab Exercises and Assignments 2.2 Dmbaker/123RF
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