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Discrete Mathematical Structures
Objectives:
Introduces the foundations of discrete mathematics as they apply to Computer Science, focusing on
providing a solid theoretical foundation for further work. Further, this course aims
to
develop
understanding and appreciation of the finite nature inherent in most Computer Science problems and
structures through study of combinatorial reasoning, abstract algebra, iterative procedures, predicate
calculus, tree and graph structures.
Reference Material:
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
Discrete Mathematics and Its Applications, by Rosen; McGraw-Hill.
Discrete Mathematics by Richard Johnsonbaugh, Prentice Hall.
Discrete Mathematical Structures, by Kolman, Busby & Ross, Prentice-Hall.
Quizzes, Lectures, Group Discussions
Reference Material:
Logic, Truth Tables, Laws of Logic
Biconditional, Argument, Application of Logic
Set Theory, Venn diagram, Set identities
Applications Of Venn diagram, Relations, Types of relations
Matrix representation of relations, Inverse of relations, Functions
Types of functions, Inverse function, Composition of functions
Sequence, Series, Recursion
Mathematical induction, Mathematical induction for divisibility
Methods of proof, Proof by contradiction, Algorithm
Division algorithm, Combinatorics, Permutations
Combinations, K-Combinations, Tree diagram
Inclusion-exclusion principle, Probability, Laws of probability
Conditional probability, Random variable, Introduction to graphs
Paths and circuits, Matrix representation of graphs, Isomorphism of graphs
Planar graphs, Trees, Spanning Trees
Week
1
2
3
4
5
Topics
Introduction/importance of Discrete Mathematics, Logic, Simple Statements, Compound
Statements, Truth Table, Basic Logic Connectives, Tautologies & Contradictions, Logic
Equivalence (DeMorgan’s Law), Principle of Substitution, Laws of the Algebra of Propositions.
Translating Word Statements to Symbolic Notation & Vice Versa, Negation of compound
statements, Arguments and their simple applications, Solving Logical Puzzles.
Set Theory: Set & its descriptive, tabular & set builder notation, Sets of Numbers (N, W, Z, E, O,
P, Q, Q*, R), Empty and Universal sets, Venn Diagram, Operations on Sets (Union, Intersection,
Difference, Complement), Venn diagram and Truth Table representation of set operations,
Subset, Set Equality, Algebra of set operations, Proving equality of two sets (using Venn
diagram, Truth Table, Laws of Algebra of Sets).
Power set, Partition of a set, Cardinality, Relations & Their Properties: Ordered pairs, Cartesian
product, Binary relation, Universal/empty relation, Arrow diagram and matrix representation of a
relation, Properties of relations, Equivalence relations.
Irreflexive and anti-symmetric relations, partially ordered relations, Composite relations, Inverse
relation, Functions: Definition of Function, Domain, Co domain and Range, Well defined
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10
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function.
Types of functions, Graph of a relation, Invertible functions, Finding inverse of a function,
Operations on functions (sum, difference, product, quotient, and composition).
Sequences (Definition and examples), Arithmetic and geometric sequences and their nth terms,
Sequences in Computer programming, Series (Definition and examples), Sum of arithmetic and
geometric series, Summation properties, Recurrence Relations: Recurrence relation (Definition
and examples), Sequences defined by recurrence relations.
Recursive definitions of: a set, union & intersection operations, sum & product, Mathematical
Induction: Principle of Mathematical Induction, Application to series sum statements, Application
to divisibility statements, Application to inequalities.
Loop Invariants: Pre conditions and Post conditions of an algorithm, Loop invariants, Loop
invariant theorem, Loop to compute a product, Correctness of division algorithm, Methods of
Proof: Converse, inverse & contra positive of a conditional statement, Direct and Indirect proofs.
Contra positive proof, Proof by contradiction, Combinatorics: Introduction, The sum and product
rules, Fictorial notation.
Counting formulas: k-samples, k-permutations, Counting formulas: k-selections, k-combinations,
Ordered partitions, Possibility tree.
Inclusion-Exclusion principle and applications, Pigeon Hole principle and applications,
Probability: Definition, Basic terminology.
Addition and Multiplication laws of probability, Conditional probability, Probability Tree, Random
variable and probability distribution, Expectation and variance.
Graph and Trees: Graph (Introduction), Basic Terminology, Types of Graphs, Matrix
Representation of Graphs, Isomorphic Graphs.
Eulerian and Hamiltonian Graphs, Trees (Definition and Examples), Characterizing Trees,
Traversing a Binary Tree.
Progress
Week
Topics
1
Introduction, Logic, Statements, Truth Table, Basic Logic Connectives, Tautologies &
Contradictions, Logic Equivalence, Principle of Substitution, Laws of the Algebra of Propositions.
2
Translating Word Statements to Symbolic Notation & Vice Versa, Negation of compound
statements, Arguments and their simple applications, Solving Logical Puzzles.
3
Sets, Notations, Sets of Numbers, Empty and Universal sets, Venn Diagram, Operations,
Representations of operations, Subset, Equality, Algebra of operations, Equality of two sets.
4
Hourly-I, Power set, Partitions, Cardinality, Relations, Ordered pairs, Cartesian product, Binary
relation, Universal/empty relation, Arrow diagram, Matrix representation, Properties,
Equivalence.
5
Irreflexive and anti-symmetric relations, partially ordered relations, Composite relations, Inverse
relation, Functions: Definition of Function, Domain, Co domain and Range, Well defined
function.
6
Types of functions, Graph of a relation, Invertible functions, Finding inverse of a function,
Operations on functions (sum, difference, product, quotient, and composition).
7
Sequences (Definition and examples), Arithmetic and geometric sequences and their nth terms,
Sequences in Computer programming, Series (Definition and examples), Sum of arithmetic and
geometric series, Summation properties, Recurrence Relations: Recurrence relation (Definition
and examples), Sequences defined by recurrence relations.
8
Recursive definitions of: a set, union & intersection operations, sum & product, Mathematical
Induction: Principle of Mathematical Induction, Application to series sum statements, Application
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10
11
12
13
14
15
to divisibility statements, Application to inequalities.
Loop Invariants: Pre conditions and Post conditions of an algorithm, Loop invariants, Loop
invariant theorem, Loop to compute a product, Correctness of division algorithm, Methods of
Proof: Converse, inverse & contra positive of a conditional statement, Direct and Indirect proofs.
Contra positive proof, Proof by contradiction, Combinatorics: Introduction, The sum and product
rules, Fictorial notation.
Counting formulas: k-samples, k-permutations, Counting formulas: k-selections, k-combinations,
Ordered partitions, Possibility tree.
Inclusion-Exclusion principle and applications, Pigeon Hole principle and applications,
Probability: Definition, Basic terminology.
Addition and Multiplication laws of probability, Conditional probability, Probability Tree, Random
variable and probability distribution, Expectation and variance.
Graph and Trees: Graph (Introduction), Basic Terminology, Types of Graphs, Matrix
Representation of Graphs, Isomorphic Graphs.
Eulerian and Hamiltonian Graphs, Trees (Definition and Examples), Characterizing Trees,
Traversing a Binary Tree.