Concept of Relation

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  • Angle and Its Measurement
    • Directed Angle
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    • Angles in Standard Position
    • Measures of Angles
    • Area of a Sector of a Circle
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  • Trigonometry - 1
    • Introduction of Trigonometry
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    • Signs of Trigonometric Functions in Different Quadrants
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    • 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
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    • Trigonometric Functions of Triple Angle
    • Factorization Formulae
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    • Formulae for Conversion of Product in to Sum Or Difference
    • Trigonometric Functions of Angles of a Triangle
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  • Probability
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    • Fundamental Principles of Counting
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    • Permutations When Repetitions Are Allowed
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    • Circular Permutations
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  • 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
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  • Limits
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    • Substitution Method
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    • Limit at Infinity
  • Continuity
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  • 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
  • Definition of Relation
  • Domain
  • Co-domain and Range of a Relation

Definition

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product  A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The second element is called the image of  the first element.

The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range ⊂ codomain.

Notes

(i) A relation may be represented algebraically either by the Roster method or by the Set-builder method. 
(ii) An arrow diagram is a visual representation of a relation.
Consider the two sets P = {a, b, c} and Q = {Ali, Bhanu, Binoy, Chandra, Divya}. 

The cartesian product of P and Q has 15 ordered pairs which can be listed as P × Q = {(a, Ali), (a,Bhanu), (a, Binoy), ..., (c, Divya)}. We can now obtain a subset of P × Q by introducing a relation R between the first element x and the second element y of each ordered pair (x, y) as R= { (x,y): x is the first letter of the name y, x ∈ P, y ∈ Q}. Then R = {(a, Ali), (b, Bhanu), (b, Binoy), (c, Chandra)} A visual representation of this relation R (called an arrow diagram) is shown in Fig .
 A relation R from A to A is also stated as a relation on A.

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