Relations and Functions-II, Objectives, Relation, Types of Relations

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  • Let and be two sets. Then a relation from Set into Set is a subset of .

  • Thus, is a relation from to

  • If then we write which is read as ‘’ is related to by the relation , if , then we write and we say that is not related to by the relation .

  • If and , then has mn ordered pairs, therefore, total number of relations form to is .

Types of Relations

(i) Reflexive Relation:

  • A relation on a set is said to be reflexive if every element of is related to itself.

  • Thus, is reflexive for all

  • A relation is not reflexive if there exists an element such that .

  • Let be a set. Then

  • is a reflexive relation on .

  • But is not a reflexive relation on , because but .

(ii) Symmetric Relation:

  • A relation on a set is said to be symmetric relation if

  • for all

  • i.e. for all .

  • Let and and be relations on given by

  • and

  • is symmetric relation on because

  • or for all

  • but is not symmetric because but .

  • A reflexive relation on a set is not necessarily symmetric. For example, the relation

  • is a reflexive relation on set but it is not symmetric.

(iii) Transitive Relation:

  • Let be any set. relation on is said to be transitive relation if

  • and for all

  • i.e. and for all

For example:

On the set of natural numbers, the relation defined by

is less than ’, is transitive, because for any


i.e. and


Check the relation for reflexivity, symmetry and transitivity, where is defined as iff for all


Let be the set of all lines in a plane. Given that for all

Reflexivity: is not reflexive because a line cannot be perpendicular to itself i.e. is not true.

Symmetry: Let such that


So, is symmetric on

Transitive: is not transitive, because and does not impty that

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