# s4_1 - CHAPTER 4 Applications of Methods of Proof 1 Set...

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CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are analogous to the operations , , and ¬ , respectively, that were defined for propositions. Indeed, each set operation was defined in terms of the corresponding operator from logic. We will discuss these operations in some detail in this section and learn methods to prove some of their basic properties. Recall that in any discussion about sets and set operations there must be a set, called a universal set , that contains all others sets to be considered. This term is a bit of a misnomer: logic prohibits the existence of a “set of all sets,” so that there is no one set that is “universal” in this sense. Thus the choice of a universal set will depend on the problem at hand, but even then it will in no way be unique. As a rule we usually choose one that is minimal to suit our needs. For example, if a discussion involves the sets { 1 , 2 , 3 , 4 } and { 2 , 4 , 6 , 8 , 10 } , we could consider our universe to be the set of natural numbers or the set of integers. On the other hand, we might be able to restrict it to the set of numbers { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } . We now restate the operations of set theory using the formal language of logic. 1.2. Equality and Containment. Definition 1.2.1 . Sets A and B are equal , denoted A = B , if x [ x A x B ] Note: This is equivalent to x [( x A x B ) ( x B x A )] . Definition 1.2.2 . Set A is contained in set B (or A is a subset of B ), denoted A B ,if x [ x A x B ] . 96

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1. SET OPERATIONS 97 The above note shows that A = B iff A B and B A. 1.3. Union and Intersection. Definitions 1.3.1 . The union of A and B , A B = { x | ( x A ) ( x B ) } . The intersection of A and B , A B = { x | ( x A ) ( x B ) } . If A B = , then A and B are said to be disjoint . 1.4. Complement. Definition 1.4.1 . The complement of A A = { x U ( x A ) } = { x U | x 6∈ A } . Discussion There are several common notations used for the complement of a set. For exam- ple, A c is often used to denote the complement of A . You may find it easier to type A c than A , and you may use this notation in your homework. 1.5. Difference. Definition 1.5.1 . The difference of A and B , or the complement of B rela- tive to A , A - B = A B. Definition 1.5.2 . The symmetric difference of A and B , A B = ( A - B ) ( B - A ) = ( A B ) - ( A B ) . Discussion The difference and symmetric difference of two sets are new operations, which were not defined in Module 1.1 . Notice that B does not have to be a subset of A for
1. SET OPERATIONS 98 the difference to be defined. This gives us another way to represent the complement of a set A ; namely, A = U - A , where U is the universal set.

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