BCA Discrete Structure

Basic Concepts: Sets, elements, roster and set-builder notation, cardinality, Set Relationships, Subsets, Proper subsets, Universal set, Complement, Disjoint sets, Set Operations, Union, Intersection, Difference, Complement, Symmetric difference, Venn Diagrams: Visual representation of set relationships and operations, Set Identities: Proof of identities using algebraic and Venn diagram methods, Cartesian Products: Ordered pairs, cross product of two or more sets, Power Sets: Definition and computation of power sets, Applications: Use of sets in databases, computer programming, and decision structures.

Propositions and Logical Operators: Definition of propositions, types (simple, compound), logical connectives: AND, OR, NOT, IMPLICATION, BICONDITIONAL, Truth Tables: Constructing truth tables for expressions involving logical operators, Tautologies, Contradictions, and Contingencies: Identifying always true/false/logical expressions, Logical Equivalence and Implications: Laws of logic (De Morgan's, distributive, associative, etc.), verifying equivalences, Predicate Logic and Quantifiers: Introduction to predicates, universal and existential quantifiers, Rules of Inference: Modus ponens, modus tollens, hypothetical syllogism, and others, Proof Methods: Direct, indirect, contradiction, contrapositive, and proof by cases.

Relations: Definition, Binary Relation, Representation, Domain, Range, Universal Relation, Void Relation, Union, Intersection, and Complement Operations on Relations, Properties of Binary Relations in a Set: Reflexive, Symmetric, Transitive, Anti-symmetric Relations, Relation Matrix and Graph of a Relation; Partition and Covering of a Set, Equivalence Relation, Equivalence Classes, Compatibility Relation, Maximum Compatibility Block, Composite Relation, Converse of a Relation, Transitive Closure of a Relation R in Set X, examples from real-world scenarios, Representation of Relations: Using matrices and directed graphs (digraphs), Equivalence and Partial Order Relations: Properties and examples, Simple or Linear Ordering, Totally Ordered Set (Chain), Frequently Used Partially Ordered Relations, Representation of Partially Ordered Sets, Hesse Diagrams, Least & Greatest Members, Minimal & Maximal Members, Least Upper Bound (Supremum), Greatest Lower Bound (Infimum), Well-ordered Partially Ordered Sets (Posets), Lattice as Posets, complete, distributive modular and complemented lattices and pseudo Boolean lattices (Definitions and simple examples only), Closures and Composition of Relations: Reflexive, symmetric, transitive closures, Functions: Definition, domain, co-domain, range, examples, Types of Functions: Injective (one-to-one), surjective (onto), bijective (one-to-one correspondence), Inverse and Composition of Functions: Definitions and computations, Applications: Use in programming, data mapping, and relational databases.

Mathematical Reasoning: Basic structure of arguments, logical flow, Mathematical Induction: Principle of induction, proof by induction, applications in series and recursive definitions, Strong Induction: Differences from regular induction, applications, Recursive Definitions: Defining sequences and structures recursively, Structural Induction: Proofs involving recursively defined structures like trees and lists, Applications: Problem-solving and validation of algorithms.

Basic Counting Principles: Introduction to counting, rule of sum and rule of product with real-world examples (e.g., menu combinations, clothing combinations), Permutations and Combinations: Concepts of ordered and unordered selections, factorial notation, formulae for permutations (nPr) and combinations (nCr), applications in password generation and team selection, Pigeonhole Principle: Understanding the concept, simple and strong pigeonhole principle, applications such as birthday paradox, drawer problems, and error checking, Inclusion-Exclusion Principle: Set-based approach to solving overlapping sets, solving problems involving counting elements in unions of sets (up to three sets), and its application in probability and combinatorics.

Graphs: Introduction, definition, examples; Nodes, edges, adjacent nodes, directed and undirected edge, Directed graph, undirected graph, examples; Initiating and terminating nodes, Loop (sling), Distinct edges, Parallel edges, Multi-graph, simple graph, weighted graphs, examples, Isolated nodes, Null graph; Isomorphic graphs, examples; Degree, Indegree, out-degree, total degree of a node, examples, Subgraphs: definition, examples; Converse (reversal or directional dual) of a digraph, examples, Path: Definition, Paths of a given graph, length of path, examples; Simple path (edge simple), elementary path (node simple), examples; Cycle (circuit), elementary cycle, examples, Reachability: Definition, geodesic, distance, examples; Properties of reachability, the triangle inequality; Reachable set of a given node, Node base, examples, Connectedness: Definition, weakly connected, strongly connected, unilaterally connected, examples; Strong, weak, and unilateral components of a graph, examples, Applications to represent Resource allocation status of an operating system, and detection and correction of deadlocks, Matrix representation of graph: Definition, Adjacency matrix, boolean (or bit) matrix, examples; Determine number of paths of length n through Adjacency matrix, examples; Path (Reachability) matrix of a graph, examples; Warshall's algorithm to produce Path matrix, Flowchart, Types of Graphs: Simple, multigraph, weighted, directed/undirected, complete, bipartite, Graph Traversal: Breadth-First Search (BFS), Depth-First Search (DFS), Trees: Trees: Definition, branch nodes, leaf (terminal) nodes, root, examples; Different representations of a tree, examples; Binary tree, m-ary tree, Full (or complete) binary tree, examples; Converting any m-ary tree to a binary tree, examples; Representation of a binary tree: Linked-list; algorithms; Applications of List structures and graphs, Tree Traversals: Inorder, preorder, postorder traversal techniques, Applications: Networking, pathfinding algorithms, compiler syntax trees, file systems.

Binary Operations: Definition and examples of binary operations on sets, Algebraic Systems: Semigroups, monoids, and groups - axioms and properties, Group Theory Basics: Identity element, inverse, associativity, examples with integers and matrices, Boolean Algebra: Basic postulates and theorems, duality, Boolean functions, Logic Circuits: Simplification of logic circuits using Boolean expressions, Applications: Automata theory, logic design, cryptography.