A2 APPENDIX Review If every element of a set A is also an element of a set B , then A is a subset of B , which is denoted A B. ⊆ If two sets A and B have the same elements, then A equals B , which is denoted A B. = For example, 1, 2, 3 1, 2, 3, 4, 5 { } { } ⊆ and 1, 2, 3 2, 3, 1 . { } { } = DEFINITION Complement of a Set If A is a set, the complement of A , denoted A, is the set consisting of all the elements in the universal set that are not in A . * Figure 1 Venn diagram Universal set A B C Finding the Complement of a Set If the universal set is U 1, 2, 3, 4, 5, 6, 7, 8, 9 { } = and if A 1, 3, 5, 7, 9 , { } = then A 2, 4, 6, 8 . { } = EXAMPLE 3 Finding the Intersection and Union of Sets Let A B 1, 3, 5, 8 , 3, 5, 7 , { } { } = = and C 2, 4, 6, 8 . { } = Find: (a) A B ∩ (b) A B ∪ (c) B A C ( ) ∩ ∪ Solution EXAMPLE 2 (a) A B 1, 3, 5, 8 3, 5, 7 3, 5 { } { } { } ∩ = ∩ = (b) A B 1, 3, 5, 8 3, 5, 7 1, 3, 5, 7, 8 { } { } { } ∪ = ∪ = (c) B A C 3, 5, 7 1, 3, 5, 8 2, 4, 6, 8 3, 5, 7 1, 2, 3, 4, 5, 6, 8 3, 5 ( ) { } { } { } [ ] { } { } { } ∩ ∪ = ∩ ∪ = ∩ = DEFINITION Intersection and Union of Two Sets If A and B are sets, the intersection of A with B , denoted A B, ∩ is the set consisting of elements that belong to both A and B . The union of A with B , denoted A B, ∪ is the set consisting of elements that belong to either A or B , or both. Now Work problem 15 Usually, in working with sets, we designate a universal set U , the set consisting of all the elements that we wish to consider. Once a universal set has been designated, we can consider elements of the universal set not found in a given set. It follows from the definition of complement that A A U ∪ = and A A . ∩ = ∅ Do you see why? Now Work problem 19 It is often helpful to draw pictures of sets. Such pictures, called Venn diagrams , represent sets as circles enclosed in a rectangle, which represents the universal set. Such diagrams often help us to visualize various relationships among sets. See Figure 1. If we know that A B, ⊆ we might use the Venn diagram in Figure 2(a). If we know that A and B have no elements in common—that is, if A B ∩ = ∅ —we might use the Venn diagram in Figure 2(b). The sets A and B in Figure 2(b) are said to be disjoint . *Some texts use the notation A′ or Ac for the complement of A . Work inside parentheses first.
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