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Paper: 1MCACCE(C) – DISCRETE MATHEMATICS

This course lays the mathematical foundation required for computer science, covering topics like set theory, logic, graph theory, algebraic structures, and recurrence relations.

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Unit-I: Set Theory

1. Introduction to Set Theory:

   • Set: A collection of distinct objects (e.g., {1, 2, 3}).
   • Finite and Infinite Sets: Countable vs. uncountable sets.

2. Set Operations:

   • Union (∪): Combines all elements from sets.
   • Intersection (∩): Common elements in sets.
   • Difference (−): Elements in one set but not in another.
   • Complement: All elements not in a set.

3. Algebra of Sets:

   • Properties like commutativity, associativity, distributivity.

4. Duality:

   • Dual expressions in set theory (e.g., De Morgan’s Laws).

5. Cartesian Product:

   • Ordered pairs of elements from two sets.

6. Relations:

   • Representation of relationships between sets.
   • Types of Relations:
     • Reflexive, Symmetric, Transitive, Equivalence Relations.

7. Partial Ordering and Lattices:

   • Partial Ordering: A set with an ordering (not necessarily total).
   • Lattices: Special partial orders where every two elements have a least upper bound and greatest lower bound.

8. Functions:

   • Mapping between sets.
   • Types: Injective, Surjective, Bijective.

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Unit-II: Graphs and Trees

1. Introduction to Graphs:

   • Graph: A set of vertices and edges connecting them.
   • Directed vs Undirected Graphs:
     • Directed: Edges have direction.
     • Undirected: Edges have no direction.

2. Special Graphs:

   • Homomorphic and Isomorphic Graphs: Graphs that are structurally identical or similar.
   • Multigraph: Graph with multiple edges between the same vertices.
   • Weighted Graph: Edges have weights.

3. Paths and Circuits:

   • Eurelian Path: Visits every edge exactly once.
   • Hamiltonian Path: Visits every vertex exactly once.

4. Planar Graphs and Euler’s Formula:

   • Planar Graph: Can be drawn without edge crossings.
   • Euler’s Formula: V - E + F = 2 (Vertices, Edges, Faces).

5. Graph Coloring:

   • Assigning colors to vertices so no two adjacent vertices share the same color.

6. Trees:

   • Spanning Tree: Subgraph that connects all vertices without cycles.
   • Binary Trees: Trees where each node has at most two children.
   • Tree traversals: Preorder, Inorder, Postorder.

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Unit-III: Propositional Logic and Boolean Algebra

1. Basic Logical Operations:

   • AND (^), OR (v), NOT (~).

2. Propositions:

   • Statements that are either true or false.

3. Tautologies and Contradictions:

   • Tautology: Always true statements.
   • Contradiction: Always false statements.

4. Validity of Arguments:

   • Using truth tables to verify logical arguments.

5. Boolean Algebra:

   • Simplification of logical expressions.
   • Truth Tables and canonical forms (Sum of Products, Product of Sums).

6. Group Theory:

   • Monoid, Semigroup, and Groups:
     • Group properties: Closure, Associativity, Identity, and Inverse.
   • Subgroups:
     • Normal subgroups and Lagrange’s Theorem.
   • Applications in computer science (e.g., cryptography).

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Unit-IV: Relations and Matrices

1. Equivalence Relations:

   • Reflexive, Symmetric, and Transitive: Key properties of equivalence relations.

2. Representations of Relations:

   • Binary matrices and digraphs.

3. Closures of Relations:

   • Reflexive, Symmetric, and Transitive Closures.

4. Operations on Relations:

   • Union, Intersection, and Composition.

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Unit-V: Recursion and Recurrence Relations

1. Recursion:

   • Process of defining functions or sequences using their previous terms.

2. Recurrence Relations:

   • Linear Recurrence Relations:
     • Solving relations like a(n) = a(n-1) + 2a(n-2).
   • Constant Coefficients:
     • Homogeneous solutions and particular solutions.

3. Generating Functions:

   • Representing sequences to solve recurrence relations.

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Key Features of the Course

• Foundational Concepts: Prepares students for advanced topics like algorithms, graph theory, and cryptography.
• Real-Life Applications: Topics like graph theory are crucial for networks, AI, and optimization problems.
• Mathematical Tools: Essential for formal problem-solving in computer science.

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