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Chapter 1

Notions from Set Theory

1.1

1.1.1

Sets and Elements, Subsets

Set, examples

1. What is set theory? 2. Explain the meaning of a. Set b. Elements c. Subset d. Proper subset e. Give examples for each 3. Explain the meaning of each mathematical statement: xS xS x S, y S, z S S = {x : x R, x > 0} X=Y X=Y XY XY XY XY XY XY XY XY , ,

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2 4. Given x is a real number and larger than -4, less than 4. Write this information in te {x : (statement)} notation. 5. Describe the difference between the Analysis class consisting of one student and the one student. 6. What is the difference between and {}? 7. How many objects does represent? How many objects does {} represent? Explain.

CHAPTER 1. NOTIONS FROM SET THEORY

1.2

Operations on Sets

1. Explain the meaning of the following operations both verbally and visually. X Y X Y CX, complement of X. Use {x : (statement)x} notation to describe CX. 2. Prove that if X S, Y S, then CX CY = C(X Y ). Hint: Show that the elements described by the LHS are the same as those described by the RHS. 3. Given S is a set of integers xS (x - 2)2 = 4 a. Write down this information in{x :(statement)} notation. b. What are the elements of this set?

1.2. OPERATIONS ON SETS 4. If X and Y are sets, what are the elements of X - Y ? 5. If X and Y are disjoint sets, what are the elements of X Y X Y 6. Prove if the statements below are correct or incorrect? X Y Z = (X Y ) Z = X (Y Z) X Y Z = (X Y ) Z = X (Y Z) X (Y Z) = (X Y )(X Z) 7. What are the similarities and differences between the operations +, × and , ? 8. Prove that if I and S are sets and if for each i I we have Xi S, then C(

iI

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Xi ) =

iI

(CXi )

9. What do we mean by an ordered pair? 10. Given two sets X and Y , what is the definition of the cartesian product of X and Y?

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CHAPTER 1. NOTIONS FROM SET THEORY

1.3

Functions

Define the following terms and state the corresponding mathematical notations function from X to Y or from X into Y map or mapping graph into onto one-to-one composition of f and g composed function function of a function one-to-one correspondence inverse function image of X under f inverse image of X under f

1.4

Finite and Infinite Sets

Define the following terms and state the corresponding mathematical notations positive integers natural numbers a finite set X the number of elements in X an infinite set a sequence of n elements in a set an n-tuple of elements of a set an infinite sequence of elements of a set function of a function one-to-one correspondence inverse function image of X under f inverse image of X under f

1.5. PROBLEMS

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1.5

Problems

Do problems 1-10 in chapter 1.

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CHAPTER 1. NOTIONS FROM SET THEORY

Chapter 2

The Real Number System

2.1 The Field Properties

1. What is the real number system? 2. What are real numbers, addition, multiplication? 3. Why do we say the real number system rather than a real number system? 4. State the field properties. 5. Use the field properties to demonstrate that they lead to the following consequences: i. In a sum or product of several real numbers, parentheses can be omitted. ii. In a sum or product of several real numbers the order of the terms is immaterial. iii. For any a, b R, the equation x + a = b has one and only one solution. iv. For any a, b R, the equation xa = b with a = 0 has one and only one solution. v. For any a R, a · 0 = 0. vi. For any a R, -(-a) = a. vii. For any nonzero a R, (a-1 )-1 = a. viii. For all a, b R, -(a + b) = (-a) + (-b). ix. For nonzero a, b (ab)-1 = a-1 b-1 . R,

x. For all a R, -a = (-1) · a. 7

8 6. Are the rational numbers and the complex numbers also fields? Justify your answer.

CHAPTER 2. THE REAL NUMBER SYSTEM

2.2

Order

1. State the order property of the real numbers system. 2. Use the order property to demonstrate that it leads to the following consequences for a, b, c, d R: i. Trichotomy One and only one of the following statements is true: a>b a=b a<b ii. Transitivity If a > b and b > c, then a > c. iii. If a > b and c d, then a + c > b + d. iv. If a > b > 0 and c d > 0, then ac > bd. v. The rules of sign for adding and multiplying real numbers hold. vi. For any a R we have a2 0, with the equlaity holding only if a = 0; more generally the sum of the squares of several elements of R is always greater than or equal to zero, with equality only if all the elements in question are zero. vii. If a > b > 0, then 1/a < 1/b. ix. The computational rules of elementary arithmetic work out as consequences of our assumptions.

2.3. THE LEAST UPPER BOUND PROPERTY

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2.3

The Least Upper Bound Property

1. Define the concepts upper bound, bounded from above, least upper bound 2. Define the Least Upper Bound Property. 3. Use the the least upper bound property to demonstrate that it leads to the following consequences: i. For any real number x, there is an integer n such that n > x. ii. For any positive real number , there exists an n such that 1/n < . iii. For any x R, there is an integer n such that n x < n + 1. iv. For any x R and positive integer N , there is an integer n such that n+1 n x< . N N v. If x R, 0, then there exists a rational number r such that |x - r| .

2.4

The Existence of Square Roots

1. Prove the proposition Every positive number has a unique positive square root.

2.5

Problems

Do problems 1-16 in chapter 2.

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