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

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Objectives

After studying this lesson, you will be able to:

After studying this lesson,

After Studying this Lesson

Relation

  • 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

and

i.e. and

Example:

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

Solution:

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

Then

So, is symmetric on

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

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