Lecture 54 of 72Week 9
Equivalence Classes and Partitions
Prof. Sukumar D · IIT Hyderabad
Credits and licence
- From
- Math Foundation, IIT Hyderabad (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- 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
- Tweens profiles and older. How courses are placed
All the lectures
- 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
- Compound Statements
- Compound Statements with Quantifiers
- Implications and their negation
- Converse and Related Implications
- Biconditionals (If and Only If)
- Contrapositive and Week 2 Review
- Proofs in Mathematics
- Direct Proof Method
- Contrapositive Proof
- Proof by Contradiction
- Proving Equivalence
- Induction & Counterexamples
- Basic Set Terminology
- Empty Set, Subsets and Power Sets
- Equality of Sets
- Describing Sets and Proving Equality
- Union and Intersection
- Complement and Difference
- Family of Sets
- Union of a Family
- Intersection of a Family
- Disjoint and Pairwise Disjoint Families
- Cartesian Product
- Worked Examples and Review
- Functions: Definition and Examples
- Functions: More Examples
- Injective Functions
- Onto Functions
- Bijections
- Review: Functions
- Functions: Graphical View
- Composition of Functions
- Inverse Functions
- Graph of the Inverse Function
- Bijections between Intervals
- Review: Composition and Inverses (W6 + W7)
- 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
- 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
- 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
- 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
- Order Relations: Definition and Examples
- Restriction, Chains, and Bounds
- Maximum, Minimum, Maximal, Minimal, and LUB/GLB
- Hasse Diagrams and Intervals
- Well-Ordering, Zorn's Lemma, and the Axiom of Choice
- The Product of a Family