18 CHAPTER R Review of Basic Concepts The set of all elements belonging to set A or to set B (or to both) is the union of the two sets, written A´B. For example, if A = 51, 3, 56 and B = 53, 5, 7, 96, then we have the following. A´B = 51, 3, 56´53, 5, 7, 96 = 51, 3, 5, 7, 96 The Venn diagram in Figure 11 shows two sets A and B. Their union, A´B, is in color. Using set-builder notation, the union of sets A and B is described as follows. A∪B = 5x∣ x{A or x{B or x{both A and B6 EXAMPLE 4 Finding Complements of Sets Let U= 51, 2, 3, 4, 5, 6, 76, A = 51, 3, 5, 76, and B = 53, 4, 66. Find each set. (a) A′ (b) B′ (c) ∅′ (d) U′ SOLUTION (a) Set A′ contains the elements of U that are not in A. Thus, A′ = 52, 4, 66. (b) B′ = 51, 2, 5, 76 (c) ∅′ = U (d) U′ = ∅ S Now Try Exercise 89. Given two sets A and B, the set of all elements belonging to both set A and set B is the intersection of the two sets, written A¨B. For example, if A = 51, 2, 4, 5, 76 and B = 52, 4, 5, 7, 9, 116, then we have the following. A¨B = 51, 2, 4, 5, 76¨52, 4, 5, 7, 9, 116 = 52, 4, 5, 76 The Venn diagram in Figure 10 shows two sets A and B. Their intersection, A¨B, is in color. Using set-builder notation, the intersection of sets A and B is described as follows. A∩B = 5x∣ x{A and x{B6 Two sets that have no elements in common are disjoint sets. If A and B are any two disjoint sets, then A¨B = ∅. For example, there are no elements common to both 550, 51, 546 and 552, 53, 55, 566, so these two sets are disjoint. 550, 51, 546¨552, 53, 55, 566 = ∅ A B A " B Figure 10 EXAMPLE 5 Finding Intersections ofTwo Sets Find each of the following. Identify any disjoint sets. (a) 59, 15, 25, 366¨515, 20, 25, 30, 356 (b) 52, 3, 4, 5, 66¨51, 2, 3, 46 (c) 51, 3, 56¨52, 4, 66 SOLUTION (a) 59, 15, 25, 366¨515, 20, 25, 30, 356 = 515, 256 The elements 15 and 25 are the only ones belonging to both sets. (b) 52, 3, 4, 5, 66¨51, 2, 3, 46 = 52, 3, 46 (c) 51, 3, 56¨52, 4, 66 = ∅ Disjoint sets S Now Try Exercises 69, 75, and 85. A B A : B Figure 11
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