21. Lecture -20
The chapter provides a comprehensive exploration of different types of relations in set theory, particularly focusing on symmetric, anti-symmetric, reflexive, irreflexive, and asymmetric relations. Various properties and the number of possible relations are systematically analyzed through logical reasoning and mathematical proofs. Key methods for establishing or disproving the existence of specific types of relations are highlighted through practical examples and exercises.
Enroll to start learning
You've not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Sections
Navigate through the learning materials and practice exercises.
What we have learnt
- Relations can be classified into symmetric, anti-symmetric, reflexive, irreflexive, and asymmetric types based on specific properties.
- The number of specific types of relations on a set can be calculated with regard to their defining properties and the available ordered pairs.
- Logical reasoning and mathematical methods are crucial for proving relationships between sets and their properties.
Key Concepts
- -- Symmetric Relation
- A relation R is symmetric if for all a and b, if a is related to b (aRb), then b is also related to a (bRa).
- -- Antisymmetric Relation
- A relation R is anti-symmetric if for all a and b, if a is related to b and b is related to a (aRb and bRa), then a must be equal to b.
- -- Reflexive Relation
- A relation R is reflexive if every element is related to itself; formally, for every element a, (a, a) is in R.
- -- Irreflexive Relation
- A relation R is irreflexive if no element is related to itself; that is, for every element a, (a, a) is not in R.
- -- Asymmetric Relation
- A relation R is asymmetric if for all a and b, if a is related to b (aRb), then b is not related to a (¬(bRa)).
Additional Learning Materials
Supplementary resources to enhance your learning experience.