Sets and Their Representations

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
• 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
• 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
• 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
• 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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