Reflexive, Symmetric, Transitive & Equivalence Relations

Overview of the Lecture

• This is the third lecture in a series about relations.
• Earlier lectures provided the concepts of domain, range, and different relation types.
• This lecture focuses on reflexive, symmetric, and transitive relations.
• Equivalence relations are also explained in this lecture.
• Understanding these four relation types is key to mastering relations.

Upcoming Batch Information

• The "m22" batch is starting on April 14th on Unacademy.
• The batch is designed for JEE (Joint Entrance Examination) aspirants.
• Courses in Physics, Chemistry, and Maths will be covered.
• The batch is tailored for students entering 12th grade or those taking a "drop year".
• Features include live classes, guided tests, structured courses, study materials, and mentorship.

Reflexive Relations

• These relations are defined on a set to itself (A to A).
• In a reflexive relation, every element in set A must be related to itself.
• For example, if A = {1, 2, 3}, a reflexive relation needs to include (1, 1), (2, 2), and (3, 3).
• Reflexive relations can include additional mappings beyond the self-mapping.
• If relation R is reflexive, (a, a) must be in R for every element a in set A.
• The total number of elements in set A impacts the possible number of reflexive relations.

Calculating the Number of Reflexive Relations

• Given set A = {1, 2, 3}, calculate the number of reflexive relations.
• The mappings (1,1), (2,2), and (3,3) are essential for the relation to be reflexive
• The mappings (1,2) etc are optional.
• There are 64 possible reflexive relations for set A.
• For total relations vs. reflexive relations:
  • Total possible relations from A to A = 2^(number of elements in A squared) = 2^(3*3) = 2^9 = 512
  • The number of non-reflexive relations is: 512 - 64 = 448

General Formula for Number of Reflexive Relations

• For a set A with n elements, the number of reflexive relations from A to A is 2^(n^2 - n).

Symmetric Relations

• Symmetric relations are defined on a set to itself.
• If (a, b) is in relation R, then (b, a) must also be in R.
• Example:
  • If A = {Munna Bhaiya, Kaalin Bhaiya, Guddu Bhaiya}, and Munna Bhaiya is related to Kaalin Bhaiya, then Kaalin Bhaiya must be related to Munna Bhaiya for the relation to be symmetric.
  • If a relation does not exist it is not a requirement
• A relation R (defined from set A to A) is symmetric if, for every (a, b) in R, (b, a) is also in R.

Examples of Symmetric Relations

• Set B = {Babuji, Makbool}
  • Relation: Babuji is related to Makbool, and Makbool is related to Babuji, which forms a symmetric relation.
• Set B = {Babuji, Makbool}
  • Relation: No element is related to any other element. A symmetric relation vacuously.
• Set C = {Guddu Bhaiya, Sweety, Munna Bhaiya}
  • Relation: Guddu Bhaiya is related to Sweety, and Sweety is related to Guddu Bhaiya, and Munna is related to Munna, is a symmetric relation.
• Set D = {Babuji, Kaalin Bhaiya, Munna Bhaiya}
  • Relation: Munna Bhaiya is related to Kaalin Bhaiya, Kaalin Bhaiya is related to Babuji, and Babuji is related to Babuji, indicates a symmetric relation.

Transitive Relations

• If (a, b) is in R and (b, c) is in R, then (a, c) must also be in R.
  • i.e. if a is related to b, and b is related to c, then a must be related to c
• Diagram: If A → B and B → C, then A → C must exist for the relation to be transitive.

Examples of Transitive Relations

• Given set A = {1, 2, 3}, the lecture analyzes multiple relations to determine transitivity:
  • R1 = {(1,1), (2,2), (2,3)} is a transitive relation.
  • R2 = {(2,2), (2,1), (1,2)} is a transitive relation.
  • R3 = {(2,3), (3,1), (1,2)} is not a transitive relation.
  • R4 = {(1,2)} is a transitive relation.
  • R5 = {(1,2), (2,2)} is a transitive relation.
  • R6 = {(1,2)} is not a transitive relation.
  • R7 = {(1,2)} is not a transitive relation.

Equivalence Relations

• Relation R (on set A) is an equivalence relation if it's reflexive, symmetric, and transitive.
• A relation must meet all three conditions.
• The next session will include practice problems focusing on these relation types.