Topics
Relations and Functions
Relations and Functions
Algebra
Inverse Trigonometric Functions
Matrices
- Concept of Matrices
- Types of Matrices
- Equality of Matrices
- Operations on Matrices> Addition and Subtraction of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Matrix Multiplication
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
- Overview of Matrices
Calculus
Determinants
Vectors and Three-dimensional Geometry
Continuity and Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions
- Derivative of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Overview of Continuity and Differentiability
Linear Programming
Probability
Applications of Derivatives
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Methods of Integration>Integration Using Trigonometric Identities
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Integrals of Some Particular Functions
- Definite Integrals
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Overview of Integrals
Sets
Applications of the Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Methods of Solving Differential Equations> Variable Separable Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Overview of Differential Equations
Vectors
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
- Types of Vectors in Algebra
- Algebra of Vector Addition
- Multiplication in Vector Algebra
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors
- Overview of Vectors
Three - Dimensional Geometry
Linear Programming
Probability
Estimated time: 7 minutes
CBSE: Class 12
Types of Relations
| Type | Definition | Small Example |
|---|---|---|
| Empty Relation | A relation in which no element of A is related to any element of (A), i.e. \[(R=\phi)\]. | A={1,2}, \[R=\phi\] |
| Universal Relation | A relation in which every element of A is related to every element of A, i.e. \[R=A \times A\]. | A = {1,2}, R = {(1,1),(1,2),(2,1),(2,2)} |
| Identity Relation | A relation in which every element is related only to itself, i.e. \[I_A={(a,a):a\in A}\]. | A={1,2}, \[I_A={(1,1),(2,2)}\] |
| Reflexive Relation | A relation R on A is reflexive if \[(a,a)\in R\] for every \[a\in A\]. | R={(1,1),(2,2),(1,2)} |
| Symmetric Relation | A relation R on A is symmetric if \[(a,b)\in R \Rightarrow (b,a)\in R\]. | R={(1,2),(2,1)} |
| Transitive Relation | A relation R on A is transitive if \[(a,b)\in R\] and \[(b,c)\in R \Rightarrow (a,c)\in R\]. | R={(1,2),(2,3),(1,3)} |
| Equivalence Relation | A relation is called an equivalence relation if it is reflexive, symmetric, and transitive. | “Is equal to” relation |
CBSE: Class 12
Key Points: Types of Relations
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Empty relation and universal relation are called trivial relations.
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Identity relation always contains all self-pairs and is always reflexive.
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To check reflexivity, see whether every (a,a) is present in the relation.
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To check symmetry, see whether every (a,b) has the reverse pair (b,a).
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To check transitivity, see whether (a,b) and (b,c) together give (a,c).
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A relation is an equivalence relation only when all three properties—reflexive, symmetric, and transitive—are satisfied together.
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