54 CHAPTER 2 Sets Example 3 Element or Subset? Determine whether the following are true or false. a) ∈3 {3, 4, 5} b) ∈ {3} {3, 4, 5} c) ∈ {3} {{3}, {4}, {5}} d) {3} {3, 4, 5} # e) 3 {3, 4, 5} # f) { } {3, 4, 5} # Solution a) ∈3 {3, 4, 5} is a true statement because 3 is an element of the set {3, 4, 5}. b) ∈ {3} {3, 4, 5} is a false statement because {3} is a set, and the set {3} is not an element of the set {3, 4, 5}. c) ∈ {3} {{3}, {4}, {5}} is a true statement because {3} is an element in the set. The elements of the set {{3}, {4}, {5}} are themselves sets. d) {3} {3, 4, 5} # is a true statement because every element of the first set is an element of the second set. e) 3 {3, 4, 5} # is a false statement because the 3 is not in braces, so it is not a set and thus cannot be a subset. The 3 is an element of the set as indicated in part (a). f) {} {3, 4, 5} # is a true statement because the empty set is a subset of every set. 7 Now try Exercise 15 Table 2.1 Number of Subsets Set Subsets Number of Subsets {} {} = 1 20 a{ } a{ } {} = 2 21 a b {, } a b {, } a b { }, { } {} = × = 4 2 2 22 a b c {, , } a b c {, , } a b a c b c {, },{, },{, } a b c { },{ },{ } {} = × × = 8 2 2 2 23 By continuing this table with larger and larger sets, we can develop a general expression for determining the number of distinct subsets that can be made from any given set. Number of Subsets How many distinct subsets can be made from a given set? The empty set has no elements and has exactly one subset, the empty set. A set with one element has two subsets. A set with two elements has four subsets. A set with three elements has eight subsets. This information is illustrated in Table 2.1.
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