CSIR-UGC - Mathematics Sciences

Elementary Set Theroy

• Sets
• Null (or Empty) Set
• Subset
• Equal Sets
• Comparable Set
• Universal Sets
• Difference Set
• Complementary Set
• Union of Sets
• Intersection of Sets
• Disjoint Set
• Ordered Pair
• Cartesian Product
• Union and Intersection of an Arbitrary Family
• Relations
• Functions
• Inverse Mapping
• Composition (Product) of Mappings
• Ordered Structure
• Intervals
• Semi Open or Semi Closed Interval
• Peano’s Postulates
• Bounded and Unbounded Sets
• Order Completeness is R
• Archimedean’s Property of Real Numbers
• Dedekind’s Form of Completeness Property
• Dedekind’s Property
• Binary Operations (or Binary Compositions) is a Set
• Sub-field
• Ordered Field
• Isomorphism of Ordered Field
• Uniqueness of Complete Ordered Field
• R-as a Complete Ordered Field
• Completeness Axiom of R
• Cantor-DedeKind Axiom
• Arithmetical Continuum
• Absolute Value of R
• Neighbourhood of a Point
• Deleted Neighbourhoods
• Interior Points of a Set
• Open Set
• Limit Points of a Set 8
• Derived Set 8
• Adherent Point
• Closure of a Set
• Closed Sets
• Order Completeness Axiom
• Similar Sets
• Equivalent Sets
• Denumerable Sets
• Dedekind Pierce
• Union and Intersection of Countable Sets
• Sequence and Subsequence
• Range
• Limit of Sequence
• Limit Point of a Sequence
• Existence of a Limit Points
• Divergent Sequence
• Bounded Sequence
• Bolzano-Weierstrass Theorem
• Limit Superior and Limit Inferior
• Monotone Sequence
• Real Valued Function
• Series
• Necessary Condition for Convergence
• Cauchy’s General Principle of
• Convergent for Series
• Positive Term Series
• Comparison Test
• Cauchy’s Root Test
• D¢ Alembert’s Ratio Test
• Cauchy’s Condensation Test
• Comparison of Ratios
• Raabe’s Test
• Logarithmic Test
• De Morgan and Bertrand’s Test
• Gauss’s Test
• Integral Test
• Maclaurin’s Integral Test
• General Principle of Convergence

Linear System

• Finite Linear Combination
• Linear Span
• Intersection of Subspace
• Union of Subspaces
• Addition of Sets
• Fundamental Property of a Linear Variety
• Trivial Linear Combination
• Linear Dependent and Independent
• Maximal Linearly Independent
• Principle of Replacement
• Hat or Chapeau Functions
• Standard Basis
• Ordered Basis
• Counting of Basis
• Linear Sum
• More about Direct Sum
• Quotient Spaces
• Deficiency (or Co-dimension)
• Linear Transformations
• Linear Extension
• Range
• Kernel (Null Space)
• Rank and Nullity
• Rank-Nullity Theorem
• Singular and Non-singular Transformation
• Inverse Linear Transformation
• Consequences of Rank-Nullity Theorem
• Isomorphic
• Fundamental Isomorphism Theorem