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Mathematics

Twelve Standard >> Relation

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Relation and its types

 

A relation between two sets A and B is a subset of their Cartesian product A × B, where each element of the relation associates an element from set A with an element from set B.

Types of Relations:

Reflexive Relation:
A relation R on a set A is reflexive if every element in A is related to itself under R.
Example: Consider the set of all students in a class. The relation "is the same age as" is reflexive because every student is of the same age as themselves.

Symmetric Relation:
A relation R on a set A is symmetric if for every (a, b) ∈ R, there exists (b, a) ∈ R.
Example: Let's consider a set of cities. The relation "is connected by a direct flight to" is symmetric. If City A is connected to City B, then City B is also connected to City A.

Transitive Relation:
A relation R on a set A is transitive if, whenever (a, b) and (b, c) belong to R, then (a, c) must also belong to R.
Example: Consider a set of people. The relation "is a grandparent of" is transitive. If Person A is a grandparent of Person B, and Person B is a grandparent of Person C, then Person A is also a grandparent of Person C.

Equivalence Relation:
An equivalence relation on a set A is reflexive, symmetric, and transitive. It divides the set into disjoint subsets called equivalence classes.
Example: The relation "has the same height as" on a set of people is an equivalence relation. It's reflexive (everyone has the same height as themselves), symmetric (if A has the same height as B, then B has the same height as A), and transitive (if A has the same height as B, and B has the same height as C, then A has the same height as C).

Partial Order Relation:
A partial order relation on a set A is reflexive, antisymmetric, and transitive. It defines a partial order among the elements.
Example: The relation "is less than or equal to" on the set of integers is a partial order relation. It's reflexive (every integer is less than or equal to itself), antisymmetric (if A ≤ B and B ≤ A, then A = B), and transitive (if A ≤ B and B ≤ C, then A ≤ C).

These are some fundamental types of relations, each with its own set of characteristics and examples. They play a crucial role in various mathematical concepts and real-world scenarios.

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