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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 3.5 Proofs in Set Theory We introduced the basic notation for sets in Section 3.3. In this section, we want to apply the tech- niques for proving statements to proofs involving sets. The two fundamental definitions associated with sets involve biconditional and conditional statements. Definition. Let A and B be sets. A equals B , denoted A = B , provided that A and B contain precisely the same elements. Symbolically, A = B provided that x [( x A ) ( x B )] . Definition. Let A and B be sets. 1. A is a subset of B , denoted A B , provided that every element of A is also an element of B . Symbolically, A B provided that x [( x A ) ( x B )] . We write A 6 B to denote that A is not a subset of B . 2. A is a proper subset of B , denoted A B , provided that A B but A 6 = B . We write A 6 B to denote A is not a proper subset of B . In both of these definitions, we usually assume the sets A and B are themselves contained in some larger set, which is called a universal set . The domain of the variable x in both definitions is then assumed to be the universal set. The universal set is often determined from the context of the discussion and the specific mathematical objects under consideration. For instance, if we are considering functions in calculus, the sets that arise may be subsets of the set of real numbers, in which case we can use R as the universal set. If we are proving results in number theory, the set of integers, Z , may be an appropriate universal set. In many cases, the universal set is left unspecified. Proving Set Inclusion Given two sets A and B , to prove that A is a subset of B , A B , we need to prove the universally quantified conditional statement x [( x A ) ( x B )] . Since this is a universal statement, we start with a proof by arbitrary element. We then need to prove the conditional propositional function. Direct Proof of A B 1. Specify the variable. Suppose x is an arbitrary element of the universal set. 2. Suppose the hypothesis is true. Suppose x is an element of A . We often omit mention of the universal set and combine steps 1 and 2 into a single statement: Suppose x is an arbitrary element of A . 3. Show that the conclusion is true. We then need to show that x is an element of B . As usual with a proof by arbitrary element, we must be careful to treat x as an arbitrary element. We must show x B using only the hypothesis x A . 95 96 Chapter 3 Proof Techniques Example. Prove: For all sets A , B , and C , if A B and B C , then A C . Solution. This has symbolic form A B C [( A B ) ( B C ) ( A C )] ....
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