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2.2 - 2-11 2.2 Set Operations 10 if a = c and b = d[Hint...

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2-11 2.2 Set Operations 10 if a = c and b = d . [ Hint: First show that {{ a } , { a , b }} = {{ c } , { c , d }} if and only if a = c and b = d .] 38. This exercise presents Russell’s paradox . Let S be the set that contains a set x if the set x does not belong to itself, so that S = { x | x / x } . a) Show the assumption that S is a member of S leads to a contradiction. b) Show the assumption that S is not a member of S leads to a contradiction. By parts (a) and (b) it follows that the set S cannot be de- fined as it was. This paradox can be avoided by restricting the types of elements that sets can have. 39. Describe a procedure for listing all the subsets of a finite set. 2.2 Set Operations Introduction Two sets can be combined in many different ways. For instance, starting with the set of mathe- matics majors at your school and the set of computer science majors at your school, we can form the set of students who are mathematics majors or computer science majors, the set of students who are joint majors in mathematics and computer science, the set of all students not majoring in mathematics, and so on. DEFINITION 1 Let A and B be sets. The union of the sets A and B , denoted by A B , is the set that contains those elements that are either in A or in B , or in both. An element x belongs to the union of the sets A and B if and only if x belongs to A or x belongs to B . This tells us that A B = { x | x A x B } . The Venn diagram shown in Figure 1 represents the union of two sets A and B . The area that represents A B is the shaded area within either the circle representing A or the circle representing B . We will give some examples of the union of sets. EXAMPLE 1 The union of the sets { 1 , 3 , 5 } and { 1 , 2 , 3 } is the set { 1 , 2 , 3 , 5 } ; that is, { 1 , 3 , 5 } ∪ { 1 , 2 , 3 } = { 1 , 2 , 3 , 5 } . EXAMPLE 2 The union of the set of all computer science majors at your school and the set of all mathematics majors at your school is the set of students at your school who are majoring either in mathematics or in computer science (or in both). DEFINITION 2 Let A and B be sets. The intersection of the sets A and B , denoted by A B , is the set containing those elements in both A and B . An element x belongs to the intersection of the sets A and B if and only if x belongs to A and x belongs to B . This tells us that A B = { x | x A x B } .
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11 2 / Basic Structures: Sets, Functions, Sequences, and Sums 2-12 U B A A B is shaded. FIGURE 1 Venn Diagram Representing the Union of A and B . U B A A B is shaded. FIGURE 2 Venn Diagram Representing the Intersection of A and B . The Venn diagram shown in Figure 2 represents the intersection of two sets A and B . The shaded area that is within both the circles representing the sets A and B is the area that represents the intersection of A and B . We give some examples of the intersection of sets.
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