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ALGEBRA AND CALCULUS

ALGEBRA AND CALCULUS


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माहिती

प्रस्तावना

1. Relations and Functions
   Introduction
1. Ordered Pairs and Cartesian Product
1.1 Cartesian Product of two sets
2. Relations, Types of Relations, Equivalence Relations, Partial Orderings
2.1 Types of Relations
2.2 Equivalence Relations
2.3 Partial Ordering
3. Equivalence Class, Properties and Partition of a Set
3.1 Equivalence Classes
3.2 Partition of set and equivalence classes
4. Transitive Closure and Warshall’s algorithm
4.1 Transitive closure
4.2 Warshall’s Algorithm
5. Digraphs of Relations, Matrix representation and Composition of               Relations
5.1 Digraphs of Relations
5.2 Matrix Representation and Composition of Relations
5.3 Composing the Relations
6. Definition of Function as Relation, Types of Function (One-one, Onto,         Bijective)
6.1 Types of Functions
2. Binary Operations and Groups
1. Definition of Binary Operation, Examples, Properties
1.1 Properties of Binary operation
2. Definition of Monoid, Semigroup, Examples
3. Definition of Group and Examples, Finite and Infinite Groups,                   Permutation Groups, Sub Groups, Cyclic Groups
3.1 Finite and Infinite Groups
3.2 Permutation Group
3.3 Cycles and Cyclic Notation
3.4 Subgroups
3.5 Cyclic Subgroup
3.6 Cyclic Groups
3. Divisibility in Integers
1. Well Ordering Principle
2. First and Second Principle of Mathematical Induction
3. Division Algorithm
4. Divisibility and it’s Properties, Prime Numbers
4.1 Divisibility - Properties
4.2 Prime Number
5. Definition of G.C.D. and L.C.M., G.C.D. as a Linear Combination of two       Integers
    Alternative Definition of gcd 21
6. Euclidean Algorithm (Without Proof)
7. Least Common Multiple
8. Relatively Prime Integers, Euclid’s Lemma and its Generalization
8.1 Relatively Prime Integers
8.2 Euclid’s Lemma
8.3 Generalization of Euclid’s Lemma
9. Congruence Relations and its Properties, Residue Classes, Definition,         Examples, Addition and Multiplication Modulo n and Composition Tables
9.1 Congruence Relation and its Properties
9.2 Residue Classes
10. Euler’s and Fermat’s Theorems (Without Proof)
10.1 Euler’s f Function
10.2 Euler’s Theorem
4. Continuity and Differentiability
1. Introduction
2. Continuity and Properties of Continuous Functions Defined on [a,b]           (Without Proof) and Examples
2.1 Elementary Properties of Continuous Functions
2.2 Properties of Continuous Functions on [a, b]
2.3 Definitions for Functions
2.4 Properties of Continuous Functions (Without Proof)
3. Differentiability
4. Theorem- Differentiability Implies Continuity but not conversely, Left         Hand Derivative and Right Hand Derivative
4.1 Right Hand and Left Hand Derivatives
4.2 Differentiability in [a, b]
4.3 Differentiability Implies Continuity
5. Intermediate Value Theorem (Without Proof)
6. Rolle’s Theorem
6.1 The Statement of Rolle’s Theorem
6.2 Geometrical Interpretation of Rolle’s Theorem
7. Lagrange’s Mean Value Theorem
7.1 Geometrical Interpretation of LMVT
8. Cauchy’s Mean Value Theorem
9. L’Hospital’s Rule (Without Proof)
10. Stronger Form
5. Successive Differentiation
1. The nth Derivatives of Standard Functions
2. Leibnitz’s Theorem (With Proof)
6. Taylor’s and Maclaurin’s Theorems
1. Taylor’s and Maclaurin’s Theorems with Lagrange’s and Cauchy’s Form     of Remainders (Without Proof)
1.1 Taylor’s Theorem with Remainder
1.2 Maclaurian’s Theorem with Remainder
2. Taylor’s and Maclaurin’s Series
7. Matrices and System of Linear Equations
1. Introduction
2. Revision: Elementary Operations on Matrices
3. Echelon Form of a Matrix
3.1 Reduction of a mxn Matrix to Row Echelon Form
4. System of Linear Equations: Gauss Elimination Method, Gauss-Jordan       Elimination Method, L.U. Decomposition Method
4.1 System of Linear Equation
4.2 Gauss Elimination Method
4.3 Gauss-Jordan Elimination Method
4.4 LU-Decomposition Method
5. Rank of a Matrix, Row Rank, Column Rank
5.1 Row Rank, Column Rank

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ALGEBRA AND CALCULUS

ALGEBRA AND CALCULUS