2.3 Venn Diagrams, Set Operations, and Data Representation 59 exactly the same elements as set B, that is, = A B, the two sets may be illustrated as in Fig. 2.2(c). Two sets A and B with some elements in common are shown in Fig. 2.2(d), which is regarded as the most general form of a Venn diagram. If we label the regions of the diagram in Fig. 2.2(d) using I, II, III, and IV, we can illustrate the four possible cases with this one diagram, Fig. 2.3. CASE 1: DISJOINT SETS When sets A and B are disjoint, they have no elements in common. Therefore, region II of Fig. 2.3 is empty. CASE 2: SUBSETS When A B, # every element of set A is also an element of set B. Thus, there can be no elements in region I of Fig. 2.3. If B A, # however, then region III of Fig. 2.3 is empty. CASE 3: EQUAL SETS When set = A B set , all the elements of set A are elements of set B and all the elements of set B are elements of set A. Thus, regions I and III of Fig. 2.3 are empty. CASE 4: OVERLAPPING SETS When sets A and B have elements in common, those elements are in region II of Fig. 2.3. The elements that belong to set A but not to set B are in region I. The elements that belong to set B but not to set A are in region III. In each of the four cases, any element belonging to the universal set but not belonging to set A or set B is placed in region IV. Next we introduce set operations. Venn diagrams will be helpful in understanding set operations. The basic operations of arithmetic are + − × , , ,and ÷. When we see these symbols, we know what procedure to follow to determine the answer. Some of the operations in set theory are ′ − , , , , > < and ×. They represent complement, intersection, union, difference, and Cartesian product, respectively. Complement U U U U A A A A B B B B (a) (c) (b) (d) Figure 2.2 U A B I II III IV Figure 2.3 U A A9 Figure 2.4 Definition: Complement The complement of set A, symbolized by ′A, is the set of all the elements in the universal set that are not in set A. Learning Catalytics Keyword: Angel-SOM-2.3 (See Preface for additional details.) Using set-builder notation, the complement of set A is indicated by ′ = ∈ A x x U { | ∉ x A and }. In Fig. 2.4, the shaded region outside set A within the universal set represents the complement of set A, or ′A. Example 1 A Set and Its Complement Given = = U abcdef g A ace {, , , , , , }and {, , } determine ′A and illustrate the relationship among sets U, A, and ′A in a Venn diagram. Solution The elements in U that are not in set A are b d f g , , , . Thus, ′ = A b d f g {, , , }. The Venn diagram is illustrated in Fig. 2.5. The elements b d f g , , , can be placed anywhere in the shaded area. 7 Now try Exercise 9 e a c b d f g U A A9 Figure 2.5
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