Q. 154.3( 4 Votes )

If R and S are relations on a set A, then prove the following :

(i) R and S are symmetric R S, and R S is symmetric

(ii) R is reflexive, and S is any relation R S is reflexive.

Answer :

R and S are two symmetric relations on set A

(i) To prove: R S is symmetric


Symmetric: For Symmetric, we need to prove that-


If (a, b) R, then (b, a) R


Let (a, b) R S


(a, b) R and (a, b) S


(b, a) R and (b, a) S


[ R and S are symmetric]


(b, a) R S


R S is symmetric


To prove: R S is symmetric


Symmetric: For Symmetric, we need to prove that-


If (a, b) R, then (b, a) R


Let (a, b) R S


(a, b) R or (a, b) S


(b, a) R or (b, a) S


[ R and S are symmetric]


(b, a) R S


R S is symmetric


(ii) R and S are two relations on a such that R is reflexive.


To prove : R S is reflexive


Reflexivity : For Reflexivity, we need to prove that-


(a, a) R


Suppose R S is not reflexive.


This means that there is a R S such that (a, a) R S


Since a R S,


a R or a S


If a R, then (a, a) R


[ R is reflexive]


(a, a) R S


Hence, R S is reflexive


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 01Functions - 01Functions - 0152 mins
Some standard real functionsSome standard real functionsSome standard real functions61 mins
Quick Revision of Types of RelationsQuick Revision of Types of RelationsQuick Revision of Types of Relations59 mins
Range of FunctionsRange of FunctionsRange of Functions58 mins
Functions - 06Functions - 06Functions - 0648 mins
Functions - 11Functions - 11Functions - 1156 mins
Battle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their Graphs48 mins
Functions - 09Functions - 09Functions - 0947 mins
Quick Recap lecture of important graphs & functionsQuick Recap lecture of important graphs & functionsQuick Recap lecture of important graphs & functions58 mins
Range of Quadratic/quadratic & linear/Linear functionsRange of Quadratic/quadratic & linear/Linear functionsRange of Quadratic/quadratic & linear/Linear functions45 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses