Course module
Discrete Mathematics — Discrete Mathematics
- Lessons
- 136
- Duration
- 10 hr 11 min
- PYQs
- 0
- Practice
- 204
Syllabus index
Lessons in this module
Set Theory
Sets, Representation of Sets and Hierarchy of numbers
Finite, Infinite, Countable, and Uncountable Sets
Null, Universal, Subsets, Proper Subsets
Power Set and its Cardinality
Set Operations – Union, Intersection, Set Difference, Symmetric Difference, and Complement
Relations
Cartesian Product, Relation, Inverse & Complement
Reflexive Irreflexive Relation – Count, and Properties
Symmetric Anti-Symmetric Asymmetric Relation and Properties
Transitive Relation – Definition, Count, and Properties
Equivalence Relation, Equivalence Classes, Partitions of a Set
Partial Order Relation – Properties, Partially Ordered Set (Poset), and Total Order Relation
Hasse Diagram, Greatest Element, Least Element, Upper Bound, Lower Bound
Lattice – Definition, Formation, and Examples of Join and Meet Operations
Bounded, Unbounded, Distributive, Complemented Lattices and Boolean Algebr
Functions
Definition of a Function – Domain, Co-domain, Range, and Count of Possible Functions
Composition of Functions – Definition, Properties, and Examples
One-to-One (Injective) Function – Definition, Count, and Properties
Onto (Surjective) Function – Definition, Count, and Properties
Bijective Function – Definition, Count, and Properties
Inverse of a Function – Definition, Existence Conditions, and Examples
Graph Theory
Types of Graphs – Finite, Infinite, Null, Trivial, Complete
Bipartite, Cycle, Regular, and Complement of a Graph
Number of Graphs – Counting of Simple, Undirected, Unlabeled
Degree of Vertex – Isolated and Pendant Vertices, Handshaking Lemma, and Degree Sequence
Minimum and Maximum Degree – Relationships, Degree Constraints, and Havel-Hakimi Theorem
Graph Traversal – Walk, Path, Trail, Circuit, and Connected Graphs
Euler and Hamiltonian Graphs – Definitions, Conditions, and Examples
Planar Graphs – Kuratowski’s Theorems, Homorphism and examples
Euler Formula – Planar Graph Formula, Applications, and Derived Relations
Graph Coloring – Vertex Coloring, Edge Coloring, Chromatic Number, and Coloring Theorems
Trees – Definitions, Properties, Eccentricity, Diameter, Radius, and Center
Spanning Tree and Spanning Forest – Definition, Construction, and Applications
Cut Set and Connectivity – Edge Connectivity, Vertex Connectivity, and Cut Set Concepts
Graph Isomorphism – Definition, Detection, and Problem Solving
Graph Matching – Maximal, Maximum, Perfect Matching, and Related Concepts
Graph Covering – Line and Vertex Covering, Independent Set, and Minimal/Maximal Variants
Group Theory
Closure-Algebraic Structure, Associative-SemiGroup, Identity-Monoid
- Basics of Group TheoryDrm Video
- Closure Property and Algebraic StructureDrm Video
- Problems on Closure Property and Algebraic StructureDrm Video
- Associative Property and Semi-GroupsDrm Video
- Problems on Associative Property and Semi-GroupsDrm Video
- Identity Property and MonoidDrm Video
- Problems on Identity Property and MonoidDrm Video
Inverse Property and Group
Commutative Property and Abelian Groups
Classification of Finite and Infinite Groups
Subgroup Definition, Examples, and Verification Techniques
Determining the Order of Elements within Groups
Propositional and Predicate Logic
Introduction to Propositions, Laws of Contradiction and Excluded Middle
Logical Operators – Negation, Conjunction, and Disjunction
Implication and Bi-Conditional Operators in Logic
Types of Logical Cases – Tautology, Contradiction, Contingency, Satisfiability, and Validity
Introduction to First Order Predicate Logic
Quantifiers – Universal and Existential
Practice Problems on Quantifiers and Predicate Logic
Functional Completeness
Questions and Practice Problems on Propositions
Multi-Topic Content (Cross-Concept Material)
Minimum Spanning Trees
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