Discrete Structures – Teaching Plan
Book: Discrete Mathematics and Its Applications, 7th Edition, K. H. Rosen.
1. Ch. 1: The Foundations: Logic and Proofs
1.1. Section 1.1: Propositional Logic
Introduction, propositions and propositional variables, truth tables, negation,
conjunction, disjunction, exclusive OR, conditional and biconditional
statements, converse, contrapositive and inverse, truth tables of compound
propositions, precedence of logical operators, bit operations. Examples 1 – 13.
Exercises 1.1. Q1 to Q40 (odd) HW.
1.2. Section 1.3: Propositional Equivalences
Introduction, tautology and contradiction, De Morgan Laws, logical
equivalences listed in Table 6, 7, 8. Examples 1 – 8. Exercises 1.3. Q1 to Q33
(odd) HW.
1.3. Section 1.4: Predicates and Quantifiers
Predicate and its domain, Examples 1 – 6. Quantifiers, the universal quantifier,
counter examples, the existential quantifiers, Examples 7 – 16. Precedence of
quantifiers, binding variables, logical equivalences involving quantifiers,
negating quantified statements, Translating from informal language to formal
language, Examples 18 – 27. Exercise 1.4 Q1 to Q40 (odd) HW.
1.4. Section 1.5: Nested Quantifiers
Introduction, quantification and loops, the order of quantifiers, translating
statements into logical expressions, negating nested quantifiers. Examples: 1 –
7, 9 – 12, 14, 15. Exercises 1.5
Q1 – Q12 HW.
1.5. Section 1.6: Rules of Inference Valid arguments in propositional logic,
rules of inference for propositional logic.
Examples 1 – 5.
2. Ch. 2: Basic Structures: Sets, Functions, Sequences, and Sums
2.1. Section 2.1: Sets
Introduction, describing sets, empty sets, equality of sets, singleton set, subsets,
Venn diagrams, power set, cartesian product, cardinality of a set, truth sets.
Examples 1 – 23. Exercise 2.1 Q1 – Q40 (odd) HW.
2.2. Section 2.2: Set Operations
Union, intersection, and difference of sets, disjoint sets, set identities,
membership tables. Examples 1 – 16. Exercise 2.2 Q1 – Q40 (odd).
2.3. Section 2.3: Functions
Introduction, definition of a function, domain, codomain and range of a
function, sum and product of functions, the image of a set, one-to-one and onto
function, bijective functions, invertible functions,
composition of functions. Examples 1 – 23. Exercises 2.3 Q1 to Q 23, Q30,
Q31, Q36, Q37 HW.
2.4. Section 2.4: Sequences and Summations
Sequences, the nth term, geometric and arithmetic progressions or sequences,
recurrence relations, the solution of recurrence relations and initial conditions,
closed or explicit formula. Examples 1 – 11.
Some useful sequences, summations, index of summation, double summation,
Examples 17 – 22.
Exercises 2.4 Q1 to Q23, Q29 to Q34 (odd) HW.
3. Ch. 5: Induction and Recursion
3.1. Section 5.1: Mathematical Induction Introduction, principle of
mathematical induction: basis step and inductive step. Template for Proofs
by Mathematical Induction, Examples 1 – 6, 8. Exercises 5.1 Q3 to Q10, Q18
to Q211, Q31 to Q34.
3.2. Section 5.3: Recursive Definition and Structural Induction
Recursively defined functions, Examples 1 – 3. Exercises 5.3 Q1 to Q8, Q27,
Q28.
3.3. Section 5.4: Recursive Algorithms
Definition of recursive algorithm, Examples 1 – 3. Exercises 5.3: Q1 to Q3.
Prepared by Dr. Rashid Ali
4. Ch. 6: Counting
4.1. Section 6.1: The Basics of Counting
The product rule and the sum rule of counting, Examples 1 – 10, 13, 14.
Exercises 6.1: Q1 to Q12, Q17, Q25, Q28, Q29, Q31, Q32, Q33, Q48.
4.2. Section 6.2: The Pigeonhole Principle The illustration of the pigeonhole
principle, the generalized pigeonhole principle. Examples 1 – 6.
Exercises 6.2: Q1 to Q6, Q13, Q15, Q17, Q19.
4.3. Section 6.3: Permutations and Combinations
Permutations, Examples 1 – 5, combinations, Examples 8 – 11. Exercises 6.3
Q1 to Q20 (odd).
5. Ch. 10: Graphs
5.1. Section 10.1: Graphs and Graph Models
Introduction, graphs, vertices, edges, end points, undirected graphs, multiple
edges, multigraphs, loops, directed graphs, pseudographs, some explanatory
examples from computer networks, graph terminology, some graph models
(Reading Assignment). Exercises 10.1 Q1 – Q10.
5.2. Section 10.2: Graph Terminology and Special Graphs
Basic terminology, neighbours, incident, neighbourhood, degree of a vertex,
isolated and pendant vertices, the handshaking theorem, initial and terminal
vertices in directed graphs, in-degree and outdegree of a vertex in directed
graphs and related results, some special simple graphs, Examples 1 – 8.
Exercises 10.2: Q1 to Q20, Q36.
5.3. Section 10.3: Representation of Graphs and Graph Isomorphism
Representing graphs with adjacency lists and adjacency matrices, isomorphisms
of graphs, incidence matrices, Examples 1 – 10. Exercises 10.3: Q1 to Q24
HW.
5.4. Section 10.4: Connectivity
Paths and related notions in undirected and directed graphs, connectedness in
undirected graphs.
Examples 1 – 4. Exercises 10.4: Q1 to Q8.
5.5. Section 10.5: Euler and Hamilton Paths
Euler path and circuits, the related results. Examples 1 – 4. Hamilton paths and
circuits, Example 5.