Chem ch 1

• Georg Cantor (1845-1918) developed the theory of sets
• G.H. Hardy: 'In these days of conflict between ancient and modern studies; there must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but is the oldest and the youngest.'
• The concept of set serves as a fundamental part of present day mathematics
• Sets are used to define the concepts of relations and functions
• The study of geometry, sequences, probability, etc. requires the knowledge of sets
• Set


  A well-defined collection of objects
• Examples of sets in mathematics

  • N: the set of all natural numbers
  • Z: the set of all integers
  • Q: the set of all rational numbers
  • R: the set of real numbers
  • Z+: the set of positive integers
  • Q+: the set of positive rational numbers
  • R+: the set of positive real numbers
• A set is a well-defined collection of objects
• Objects, elements and members of a set

  Synonymous terms
• Sets are denoted by capital letters


  A, B, C, X, Y, Z, etc.
• Elements of a set are denoted by small letters


  a, b, c, x, y, z, etc.
• a ∈ A


  a belongs to set A
• b ∉ A


  b does not belong to set A
• Methods of representing a set

  • Roster or tabular form
  • Set-builder form
• Representing sets in roster form

  • The set of all even positive integers less than 7: {2, 4, 6}
  • The set of all natural numbers which divide 42: {1, 2, 3, 6, 7, 14, 21, 42}
  • The set of all vowels in the English alphabet: {a, e, i, o, u}
  • The set of odd natural numbers: {1, 3, 5, ...}
• Set-builder form


  All the elements of a set possess a single common property which is not possessed by any element outside the set
• Representing sets in set-builder form

  • V = {x : x is a vowel in English alphabet}
  • A = {x : x is a natural number and 3 < x < 10}
  • B = {y : y is a vowel in the English alphabet}
  • C = {z : z is an odd natural number}
• Representing solutions of equations in roster form


  • The solution set of x^2 + x - 2 = 0: {1, -2}
• Representing sets in roster form

  • The set {x : x is a positive integer and x^2 < 40}: {1, 2, 3, 4, 5, 6}
• Representing sets in set-builder form

  • A = {x : x is the square of a natural number}
  • A = {x : x = n^2, where n ∈ N}
• Representing sets in set-builder form

  • {x : x = 1/n, where n is a natural number and 1 ≤ n ≤ 6}
• The order in which elements are listed in roster form is immaterial
• (i)


  {x : x is a prime number and a divisor of 6}
• (ii)


  {x : x is an odd natural number less than 10}
• (iii)


  {x : x is a natural number and divisor of 6}
• (iv)


  {x : x is a letter of the word MATHEMATICS}
• The Empty Set


  A set that does not contain any element
• A = { x : x is a student of Class XI presently studying in a school }
• B = { x : x is a student presently studying in both Classes X and XI }
• Finite set


  A set which is empty or consists of a definite number of elements
• Infinite set


  A set which is not finite
• The set of natural numbers is an infinite set
• The set of real numbers cannot be described in the roster form
• {x : x ∈ N and (x - 1)(x - 2) = 0} is a finite set
• {x : x ∈ N and x^2 = 4} is a finite set
• {x : x ∈ N and 2x - 1 = 0} is a finite set
• {x : x ∈ N and x is prime} is an infinite set
• {x : x ∈ N and x is odd} is an infinite set
• Equal sets

  Two sets A and B are equal if they have exactly the same elements
• A = {0}, B = {x : x > 15 and x < 5}, C = {x : x - 5 = 0}, D = {x: x^2 = 25}, E = {x : x is an integral positive root of the equation x^2 - 2x - 15 = 0}