Bridge to Abstract Mathematics: Mathematical Proof and Structures

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Sets CHAPTER 1 The purpose of this chapter is twofold: to provide an introduction to, or review of, the terminology, notation, a ...

4 SETS Chapter 1 theorems of set theory, as suggested by the various types of evidence you encounter. Basic Definitions and Nota ...

1.1 BASIC DEFINITIONS AND NOTATION 5 Two important facts are: (1) the order in which elements are listed is irrelevant and (2) a ...

6 SETS Chapter 1 and are not, in the set. On the other hand, if the set to be described is large, the roster method can be impra ...

1.1 BASIC DEFINITIONS AND NOTATION 7 we pointed out that Ohio c$ B = (Massachusetts, Michigan, California} but did not explicitl ...

8 SETS Chapter 1 known as intervals, which we will frequently encounter. Intervals are de- scribed by widely used notation, whic ...

1.1 BASIC DEFINITIONS AND NOTATION 9 Solution to x I (b) 7x - 9 5 16 is equivalent to 7x I 25, which is equivalent The set of so ...

10 SETS Chapter 1 Solution By completing the square, we can express 5x2 + 3x + 2 in the form 5(x + + (g), which is clearly posit ...

1.1 BASIC DEFINITIONS AND NOTATION 11 Subset. Earlier we encountered a relationship of containment between an object and a set. ...

12 SETS Chapter 1 Since "element hood" and "subset" are both relationships of containment, it is important not to confuse the tw ...

1.1 BASIC DEFINITIONS AND NOTATION 13 POWER SET We suggested earlier that sets can themselves be elements of sets. One sit- uati ...

14 SETS Chapter 1 B = {x E Rlsin zx = 0) C= {x€R(x2= -1) (d) D={x~RIx~-5x+7<0) E = {XE RI10x2 - 7x - 12 20) (f) F = {x~R1(6x ...

1.2 OPERATIONS ON SETS 15 Given the following collections of sets, find in each of parts (a), (b), and (c) all relationships of ...

16 SETS Chapter 1 Just as there is an "algebra of numbers" based on operations such as addition and multiplication, there is als ...

1.2 OPERATIONS ON SETS 17 solution to Example 1 lists numbers in increasing order, this is not neces- sary. Observe also that th ...

18 SETS Chapter 1 simplifies to "either x I -4 or x 2 1." The solution set is most con- veniently expressed (- co, -41 u [I, co) ...

1.2 OPERATIONS ON SETS 19 What do the last two parts of Example 4 suggest about the complement of the complement of a set? Also, ...

20 SETS Chapter 1 EXAMPLE 6 Let A = {1,3,5,7), B = {1,2,4,8,16), and C = {1,2,3,... , 100). Is 1 an element of A - B? Compute A ...

1.2 OPERATIONS ON SETS 21 in B, or in B but not in A. Stated differently, elements of A A B are objects in one or the other of t ...