BSc: Logic and Discrete Mathematics

Logic and Discrete Mathematics (Philosophy 1)

• Course name: Logic and Discrete Mathematics (Philosophy 1)
• Code discipline: CSE113
• Subject area: Math. Computer Science

Short Description

This course consists of two distinct but overlapping parts: i. Logic; and ii. Discrete Mathematics. The first part of the course is an introduction to formal symbolic logic. Philosopher John Locke once wrote that ``logic is the anatomy of thought. This part of the course will teach students to analyse and evaluate arguments using the formal techniques of modern symbolic logic. The second part of the is designed for students to teach them basic notions of graph theory, discrete optimization and dynamic programming. This part will give practical experience with basic algorithms in discrete mathematics.

Course Topics

Course Sections and Topics

Section	Topics within the section
Basic of Logic	Logical Operators Truth Tables for Propositions Propositional calculus Quantifiers Predicate Logic Basic Proof Techniques in Math Fundamental proofs Use of propositional calculus to do proofs
Set Theory & Finite combinatorics	Fundamentals of the set theory From sets to relations Functions and numbers Algebra of binary relations Principles of finite combinatorics Recursion and Discrete Optimization Linear recurrence relations
Basic of Graphs	From set and relations to graphs Euler tours and graphs Hamilton paths and graphs Planar graphs and Euler formula Trees and Kőnig's infinity lemma

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

This calculus course will provide an opportunity for participants to:

• understand key principles involved in differentiation and integration of functions
• solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities
• become familiar with the fundamental theorems of Calculus
• get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation.

ILOs defined at three levels

We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

• remember how to use Venn diagrams
• remember how to calculate truth values
• know the formal structures of some arguments
• explain the difference between deduction and induction
• remember how to build spanning trees
• remember how to find Euler tour and Hamilton path
• explain how to use Dijkstra’s algorithm
• know how to solve maximum flow problem

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

• apply the derivatives to analyse the functions
• integrate
• understand the basics of approximation

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

• Take derivatives of various type functions and of various orders
• Integrate
• Apply definite integral
• Expand functions into Taylor series
• Apply convergence tests

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	90-100	-
B. Good	75-89	-
C. Satisfactory	60-74	-
D. Fail	0-59	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Midterm	20
Quizzes	28 (2 for each)
Final exam	50
In-class participation	7 (including 5 extras)

Recommendations for students on how to succeed in the course

• Participation is important. Attending lectures is the key to success in this course.
• Review lecture materials before classes to do well.
• Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.

Resources, literature and reference materials

Open access resources

• Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985
• Zorich, V. A. Mathematical Analysis I, Translator: Cooke R. (2004)

Software and tools used within the course

• No.

Activities and Teaching Methods

Teaching and Learning Methods within each section

Teaching Techniques	Section 1	Section 2	Section 3
Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution)	1	1	1
Project-based learning (students work on a project)	0	0	0
Modular learning (facilitated self-study)	0	0	0
Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level)	1	1	1
Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them)	0	0	0
Business game (learn by playing a game that incorporates the principles of the material covered within the course)	0	0	0
Inquiry-based learning	0	0	0
Just-in-time teaching	0	0	0
Process oriented guided inquiry learning (POGIL)	0	0	0
Studio-based learning	0	0	0
Universal design for learning	0	0	0
Task-based learning	0	0	0

Activities within each section

Learning Activities	Section 1	Section 2	Section 3
Lectures	1	1	1
Interactive Lectures	1	1	1
Lab exercises	1	1	1
Experiments	0	0	0
Modeling	0	0	0
Cases studies	0	0	0
Development of individual parts of software product code	0	0	0
Individual Projects	0	0	0
Group projects	0	0	0
Flipped classroom	0	0	0
Quizzes (written or computer based)	1	1	1
Peer Review	0	0	0
Discussions	1	1	1
Presentations by students	0	0	0
Written reports	0	0	0
Simulations and role-plays	0	0	0
Essays	0	0	0
Oral Reports	0	0	0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

1. A plane curve is given by $x(t)=-{\frac {t^{2}+4t+8}{t+2}}$, ${\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}$. Find

   the asymptotes of this curve;

   the derivative ${\textstyle y'_{x}}$.

2. Apply Leibniz formula Find ${\textstyle y^{(n)}(x)}$ if ${\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}$.

   Draw graphs of functions

   Find asymptotes

3. Find the derivatives of the following functions:
   • ${\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}$;
   • ${\textstyle y(x)}$ that is given implicitly by ${\textstyle x^{3}+5xy+y^{3}=0}$.

Section 2

1. Find the following integrals:
   • ${\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}$;
   • ${\textstyle \int 2^{2x}e^{x}\,dx}$;
   • ${\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}$.
2. Find the indefinite integral ${\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}$.
3. Find the length of a curve given by ${\textstyle y=\ln \sin x}$, ${\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}$.

Section 3

1. Find limits of the following sequences or prove that they do not exist:
   • $a_{n}=n-{\sqrt {n^{2}-70n+1400}}$;
   • ${\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}$;
   • ${\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}$.

Final assessment

Section 1

1. Apply the appropriate differentiation technique to a given problem.
2. Find a derivative of a function
3. Apply Leibniz formula
4. Draw graphs of functions
5. Find asymptotes of a parametric function

Section 2

1. Apply the appropriate integration technique to the given problem
2. Find the value of the devinite integral
3. Calculate the area of the domain or the length of the curve

Section 3

1. Find a limit of a sequence
2. Find a limit of a function

The retake exam

Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.