Course StructureBCA 151
Discrete Structure
1
Set Theory
Sets, subsets, power sets, Cartesian products, set operations, Venn diagrams, and set identities.
Sets and Fundamentals
- Definition and representation of sets
- Roster method and set-builder notation
- Finite and infinite sets
- Empty set, singleton set, universal set
- Equal sets and equivalent sets
- Cardinal number of a set
Subsets and Power Sets
- Subsets and proper subsets
- Number of subsets of a finite set
- Power set and its cardinality
- Cartesian product of sets
- Venn diagrams and applications
Set Operations
- Union and intersection of sets
- Difference and complement of sets
- Symmetric difference
- Disjoint sets
- Properties of set operations
Set Identities and Applications
- De Morgan's laws
- Laws of complementation
- Associative and distributive laws
- Proof of set identities using membership tables
- Applications in real-world problems
2
Logic and Propositional Calculus
Propositions, logical operators, truth tables, tautologies, logical equivalence, predicates, quantifiers, and proof methods.
Propositional Logic
- Propositions and truth values
- Atomic and compound propositions
- Negation, conjunction, disjunction
- Implication and biconditional
- Conditional and converse statements
Truth Tables and Logical Equivalence
- Constructing truth tables
- Tautologies and contradictions
- Logical equivalence
- Laws of logic
- De Morgan's laws for logic
Predicates and Quantifiers
- Predicate logic
- Universal quantifier
- Existential quantifier
- Negation of quantified statements
- Nested quantifiers
Methods of Proof
- Direct proof
- Proof by contradiction
- Proof by contrapositive
- Proof by cases
- Existence and uniqueness proofs
3
Relations and Functions
Binary relations, equivalence relations, partial orders, functions, injective/surjective/bijective, composition, and inverse.
Relations
- Binary relations on sets
- Domain and range of relations
- Representation of relations (matrices, digraphs)
- Properties of relations (reflexive, symmetric, transitive)
- Equivalence relations and equivalence classes
- Partial orders and Hasse diagrams
Functions
- Definition and notation of functions
- Domain, codomain, and range
- Types of functions (injective, surjective, bijective)
- Composition of functions
- Inverse functions
- Identity functions
Special Functions
- Floor and ceiling functions
- Characteristic functions
- Permutations as bijective functions
- Growth of functions (Big-O notation)
- Pigeonhole principle and its applications
4
Mathematical Reasoning and Proof Techniques
Principle of induction, strong induction, recursive definitions, recursive algorithms, direct proof, contradiction, contrapositive, and proof by cases.
Mathematical Induction
- Principle of mathematical induction
- Base case and inductive step
- Examples using induction
- Strong induction
- Structural induction
Recursive Definitions
- Recursively defined functions
- Recursively defined sets
- Recursive algorithms
- Fibonacci sequence and recurrence relations
- Solving simple recurrence relations
Proof Techniques
- Direct proof method
- Proof by contradiction
- Proof by contraposition
- Proof by exhaustion
- Counterexamples
5
Combinatorics and Counting Principles
Basic counting rules, pigeonhole principle, permutations, combinations, binomial coefficients, and recurrence relations.
Basic Counting Principles
- Sum rule (addition principle)
- Product rule (multiplication principle)
- Inclusion-exclusion principle
- Pigeonhole principle
- Dirichlet's box principle
Permutations and Combinations
- Permutations of distinct objects
- Circular permutations
- Permutations with repetitions
- Combinations and binomial coefficients
- Pascal's identity and Pascal's triangle
Binomial Theorem
- Binomial theorem statement
- Proof by induction
- Applications in expansions
- Finding specific terms
- Multinomial theorem
Recurrence Relations
- Definition and examples
- Linear recurrence relations
- Homogeneous recurrence relations
- Particular solutions
- Solving recurrence relations
6
Graph Theory and Trees
Graphs, subgraphs, paths, cycles, connectivity, trees, spanning trees, traversals, and graph representations.
Graph Theory Fundamentals
- Definition of graphs and multigraphs
- Directed and undirected graphs
- Degree of a vertex
- Handshaking lemma
- Complete graphs, bipartite graphs
- Subgraphs and isomorphism
Paths and Circuits
- Walks, trails, and paths
- Connected and disconnected graphs
- Euler paths and Euler circuits
- Hamilton paths and Hamilton circuits
- Shortest path problem
Trees
- Definition and properties of trees
- Rooted trees and terminology
- Spanning trees
- Minimum spanning trees (Kruskyal's, Prim's)
- Tree traversals (preorder, inorder, postorder)
Graph Representations
- Adjacency matrix
- Incidence matrix
- Adjacency list representation
- Graph coloring
- Planar graphs
7
Algebraic Structures
Groups, subgroups, cosets, Lagrange's theorem, rings, fields, Boolean algebra, and lattices.
Groups
- Definition of a group
- Properties of groups
- Finite and infinite groups
- Subgroups and their tests
- Cyclic groups
Cosets and Lagrange's Theorem
- Definition of cosets
- Properties of cosets
- Lagrange's theorem
- Normal subgroups
- Quotient groups
Rings and Fields
- Definition of rings
- Properties of rings
- Subrings
- Ideals
- Fields and their properties
Boolean Algebra
- Definition of Boolean algebra
- Boolean functions
- Logic gates and circuits
- Karnaugh maps
- Simplification of Boolean expressions