Course11110600372 lectures
Math Foundation
Prof. Sukumar D · IIT Hyderabad
About
This is an introductory course to the UG students across the disciplines which needs mathematical maturity. The course is carved out by considering the necessary notions and concepts that are needed for understanding, appreciating and enjoying mathematics. It is also observed that, the clarity and confidence on these foundational concepts, empowers avoiding mis-understating or difficult in understanding. Broadly the following topics or covered: Logic (mathematical statement), Sets (both understanding and describing), Functions (interaction and interconnection among the sets both qualitatively and quantitatively), Relation on sets (mainly equivalence relation and partial orders). Proof techniques, geometric intuition, writing skills will be acquired on the way systematically. Learning them and digesting them slowly gives confidence in self-learning as well as in self-correcting.
Intended audience: Anyone want to pursue Mathematics
Prerequisites: No pre-requisite. Class 12 or equivalent.
Lectures
Week 1
- Mathematical Statements
- Negation of statements
- Universal and existential quantifiers
- Negation rules to quantified statements
- Nested quantifiers and their negation rules
- Review of statements, negation, and quantifiers
Week 2
- Compound Statements
- Compound Statements with Quantifiers
- Implications and their negation
- Converse and Related Implications
- Biconditionals (If and Only If)
- Contrapositive and Week 2 Review
Week 3
- Proofs in Mathematics
- Direct Proof Method
- Contrapositive Proof
- Proof by Contradiction
- Proving Equivalence
- Induction & Counterexamples
Week 4
- Basic Set Terminology
- Empty Set, Subsets and Power Sets
- Equality of Sets
- Describing Sets and Proving Equality
- Union and Intersection
- Complement and Difference
Week 5
- Family of Sets
- Union of a Family
- Intersection of a Family
- Disjoint and Pairwise Disjoint Families
- Cartesian Product
- Worked Examples and Review
Week 6
- Functions: Definition and Examples
- Functions: More Examples
- Injective Functions
- Onto Functions
- Bijections
- Review: Functions
Week 7
- Functions: Graphical View
- Composition of Functions
- Inverse Functions
- Graph of the Inverse Function
- Bijections between Intervals
- Review: Composition and Inverses (W6 + W7)
Week 8
- Cantor’s Theorem
- Image of Subsets under Functions — 1
- Image of Subsets under Functions — 2
- Inverse Image of Subsets under Functions
- Inverse Image of Subsets under Functions - 2
- Review: Images and Inverse Images
Week 9
- Relations: Definition and Examples
- Properties of Relations I - The Four Properties
- Properties of Relations II - More Examples, Digraphs, and Matrices
- Equivalence Relations
- Equivalence Classes and Partitions
- Equivalence Classes and Partitions
Week 10
- The Induction Principle
- The Strong Induction Principle
- The Well-Ordering Principle and the Division Algorithm
- Equivalence of the Three Principles
- Worked Examples: gcd and Bezout
- Review: Pitfalls and the Whole Picture
Week 11
- Cardinality: Comparing Sizes via Bijections
- The Schroder-Bernstein Theorem
- Knaster-Tarski and the Proof of Schroder-Bernstein
- Finite Sets and the Pigeonhole Principle
- Countable and Uncountable Sets
- Comparing Cardinality
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Hyderabad
- Department
- Mathematics
- Teacher
- Prof. Sukumar D
- Level
- Undergraduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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