3 The logical symbol is the same as if and only if If A and B are any two

3 the logical symbol is the same as if and only if if

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3 The logical symbol () is the same as “if, and only if.” If A and B are any two statements, then A () B is the same as saying A implies B and B implies A . It is also common to use iff in this way. 4 Augustus De Morgan (1806–1871) December 17, 2017 ª lee/ira
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4. FUNCTIONS AND RELATIONS 1-5 Two of the basic binary operations can be extended to work with indexed col- lections. In particular, using the indexed collection from the previous paragraph, we define [ 2 § A = { x : x 2 A for some 2 § } and \ 2 § A = { x : x 2 A for all 2 § }. De Morgan’s Laws can be generalized to indexed collections. T HEOREM 1.4. If { B : 2 § } is an indexed collection of sets and A is a set, then A \ [ 2 § B = \ 2 § ( A \ B ) and A \ \ 2 § B = [ 2 § ( A \ B ). P ROOF . The proof of this theorem is Exercise 1.4 . 4. Functions and Relations 4.1. Tuples. When listing the elements of a set, the order in which they are listed is unimportant; e.g., { e , l , v , i , s } = { l , i , v , e , s } . If the order in which n items are listed is important, the list is called an n -tuple . (Strictly speaking, an n -tuple is not a set.) We denote an n -tuple by enclosing the ordered list in parentheses. For example, if x 1 , x 2 , x 3 , x 4 are four items, the 4-tuple ( x 1 , x 2 , x 3 , x 4 ) is different from the 4-tuple ( x 2 , x 1 , x 3 , x 4 ). Because they are used so often, the cases when n = 2 and n = 3 have special names: 2-tuples are called ordered pairs and a 3-tuple is called an ordered triple . D EFINITION 1.5. Let A and B be sets. The set of all ordered pairs A £ B = {( a , b ) : a 2 A ^ b 2 B } is called the Cartesian product of A and B . 5 E XAMPLE 1.2. If A = { a , b , c } and B = {1,2}, then A £ B = {( a ,1),( a ,2),( b ,1),( b ,2),( c ,1),( c ,2)}. and B £ A = {(1, a ),(1, b ),(1, c ),(2, a ),(2, b ),(2, c )}. Notice that A £ B 6 = B £ A because of the importance of order in the ordered pairs. A useful way to visualize the Cartesian product of two sets is as a table. The Cartesian product A £ B from Example 1.2 is listed as the entries of the following table. £ 1 2 a ( a ,1) ( a ,2) b ( b ,1) ( b ,2) c ( c ,1) ( c ,2) 5 René Descartes, 1596–1650 December 17, 2017 ª lee/ira
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1-6 CHAPTER 1. BASIC IDEAS Of course, the common Cartesian plane from your analytic geometry course is nothing more than a generalization of this idea of listing the elements of a Cartesian product as a table. The definition of Cartesian product can be extended to the case of more than two sets. If { A 1 , A 2 , ··· , A n } are sets, then A 1 £ A 2 £ ··· £ A n = {( a 1 , a 2 , ··· , a n ) : a k 2 A k for 1 k n } is a set of n -tuples. This is often written as n Y k = 1 A k = A 1 £ A 2 £ ··· £ A n . 4.2. Relations. D EFINITION 1.6. If A and B are sets, then any R Ω A £ B is a relation from A to B . If ( a , b ) 2 R , we write aRb . In this case, dom( R ) = { a : ( a , b ) 2 R } Ω A is the domain of R and ran( R ) = { b : ( a , b ) 2 R } Ω B is the range of R . It may happen that dom( R ) and ran( R ) are proper subsets of A and B , respectively.
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