Sets and Their Representations

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Topics

  • Angle and Its Measurement
    • Directed Angle
    • Angles of Different Measurements
    • Angles in Standard Position
    • Measures of Angles
    • Area of a Sector of a Circle
    • Length of an Arc of a Circle
  • Trigonometry - 1
    • Introduction of Trigonometry
    • Trigonometric Functions with the Help of a Circle
    • Signs of Trigonometric Functions in Different Quadrants
    • Range of Cosθ and Sinθ
    • Trigonometric Functions of Specific Angles
    • Trigonometric Functions of Negative Angles
    • Fundamental Identities
    • Periodicity of Trigonometric Functions
    • Domain and Range of Trigonometric Functions
    • Graphs of Trigonometric Functions
    • Polar Co-ordinate System
  • Trigonometry - 2
    • Trigonometric Functions of Sum and Difference of Angles
    • Trigonometric Functions of Allied Angels
    • Trigonometric Functions of Multiple Angles
    • Trigonometric Functions of Double Angles
    • Trigonometric Functions of Triple Angle
    • Factorization Formulae
    • Formulae for Conversion of Sum Or Difference into Product
    • Formulae for Conversion of Product in to Sum Or Difference
    • Trigonometric Functions of Angles of a Triangle
  • Determinants and Matrices
  • Straight Line
    • Locus of a Points in a Co-ordinate Plane
    • Straight Lines
    • Equations of Line in Different Forms
    • General Form of Equation of a Line
    • Family of Lines
  • Circle
  • Conic Sections
    • Double Cone
    • Conic Sections
    • Parabola
    • Ellipse
    • Hyperbola
  • Measures of Dispersion
    • Meaning and Definition of Dispersion
    • Measures of Dispersion
    • Range of Data
    • Variance
    • Standard Deviation
    • Change of Origin and Scale of Variance and Standard Deviation
    • Standard Deviation for Combined Data
    • Coefficient of Variation
  • Probability
  • Complex Numbers
  • Sequences and Series
  • Permutations and Combination
    • Fundamental Principles of Counting
    • Invariance Principle
    • Factorial Notation
    • Permutations
    • Permutations When All Objects Are Distinct
    • Permutations When Repetitions Are Allowed
    • Permutations When Some Objects Are Identical
    • Circular Permutations
    • Properties of Permutations
    • Combination
    • Properties of Combinations
  • Methods of Induction and Binomial Theorem
    • Principle of Mathematical Induction
    • Binomial Theorem for Positive Integral Index
    • General Term in Expansion of (a + b)n
    • Middle term(s) in the expansion of (a + b)n
    • Binomial Theorem for Negative Index Or Fraction
    • Binomial Coefficients
  • Sets and Relations
  • Functions
  • Limits
    • Concept of Limits
    • Factorization Method
    • Rationalization Method
    • Limits of Trigonometric Functions
    • Substitution Method
    • Limits of Exponential and Logarithmic Functions
    • Limit at Infinity
  • Continuity
    • Continuous and Discontinuous Functions
  • Differentiation
    • Definition of Derivative and Differentiability
    • Rules of Differentiation (Without Proof)
    • Derivative of Algebraic Functions
    • Derivatives of Trigonometric Functions
    • Derivative of Logarithmic Functions
    • Derivatives of Exponential Functions
    • L' Hospital'S Theorem
  • Roster or Tabular method or List method
  • Set-Builder or Rule Method

Notes

Set is a collection of well defined objects. By well defined objects, we mean the definition should not vary it  should be definite.
Example 1- Collection of vowels of English alphabet. The collection will be a, e, i, o, u. This is a well defined collection.
Example 2- Collection of good maths books available in market. Here, the collection will vary form person to person because of different personal choices and preferences.
There are two methods of representing a set : 
(i) Roster or tabular form
(ii) Set-builder form.
Example- Collection of vowels of English alphabet.
A= {a, e, i, o, u} this is the roster form of a set.
Here, A is represented as a set, and the elements of set are enclosed in { }.
Set builder form, A= {x: x is a vowel in English alphabet}
x represents the elements in the enclosed brackets { }.
Note:
1) Order is immaterial. That the sequence in which the elements appear is not important, even if the sequence is changed the meaning remains  same.
Take the same example- Collection of vowels of English alphabet.
A= {e, i, o, u, a}
2) Same elements are not repeated.
Example- B= set of all letters of the word SCHOOL
Roster form: B= {S, C, H, O, L}
Set-builder form: B= {x:x is letter used in word SCHOOL}
3) C= Set of all natural numbers.
Set-builder form C = {x :x ∈ N}
Roster form C= {1, 2, 3, 4, ...}
As natural numbers are infinite so we will represent them with three dots.
∈ means belongs to and ∉ means does not belongs to.
Example- A= {1, 2, 3, 4, 5, 6}
Here, 2 ∈ A, 9 ∉ A, 8 ∉ A, 5 ∈ A
We give below a few more examples of sets used particularly in mathematics, viz. 
`"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, and 
`"R"^+` : the set of positive real numbers.

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