`Chapter 1Notions from Set Theory1.11.1.1Sets and Elements, SubsetsSet, examples1. 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 &gt; 0}  X=Y  X=Y  XY  XY  XY  XY  XY  XY  XY  XY  , ,              1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 THEORY1.2Operations on Sets1. 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(iI3Xi ) =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?4CHAPTER 1. NOTIONS FROM SET THEORY1.3FunctionsDefine 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 f1.4Finite and Infinite SetsDefine 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. PROBLEMS51.5ProblemsDo problems 1-10 in chapter 1.6CHAPTER 1. NOTIONS FROM SET THEORYChapter 2The Real Number System2.1 The Field Properties1. 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 SYSTEM2.2Order1. 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&gt;b a=b a&lt;b ii. Transitivity If a &gt; b and b &gt; c, then a &gt; c. iii. If a &gt; b and c  d, then a + c &gt; b + d. iv. If a &gt; b &gt; 0 and c  d &gt; 0, then ac &gt; 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 &gt; b &gt; 0, then 1/a &lt; 1/b. ix. The computational rules of elementary arithmetic work out as consequences of our assumptions.2.3. THE LEAST UPPER BOUND PROPERTY92.3The Least Upper Bound Property1. 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 &gt; x. ii. For any positive real number , there exists an n such that 1/n &lt; . iii. For any x  R, there is an integer n such that n  x &lt; n + 1. iv. For any x  R and positive integer N , there is an integer n such that n+1 n x&lt; . N N v. If x  R,  0, then there exists a rational number r such that |x - r|  .2.4The Existence of Square Roots1. Prove the proposition Every positive number has a unique positive square root.2.5ProblemsDo problems 1-16 in chapter 2.`

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