Q. 1E4.3( 92 Votes )

Relation R in the set A of human beings in a town at a particular time given by

A. R = {(x, y) : x and y work at the same place}

B. R = {(x, y) : x and y live in the same locality}

C. R = {(x, y) : x is exactly 7 cm taller than y}

D. R = {(x, y) : x is wife of y}

E. R = {(x, y) : x is father of y}

Answer :

(a) It is given that R = {(x, y) : x and y work at the same place}

(x, x) ϵ R


R is reflexive.


Now, if (x, y) ϵ R, then x and y work on the same place.


y and x work at the same place.


(y, x) ϵ R


R is symmetric.


Now, let (x, y), (y, z) ϵ R


x and y work at the same place and y and z work at the same place.


x and z work at the same place


(x, z) ϵ R


R is transitive.


Therefore, R is reflexive, symmetric and transitive.


(b) It is given that R = {(x, y) : x and y live in the same locality}


(x,x) ϵ R as x and x live in the same human being.


R is reflexive.


Now, if (x,y) ϵ R, then x and y live in the same locality.


y and x live in the same locality.


(y,x) ϵ R


R is symmetric.


Now, let (x,y), (y,z) ϵ R


x and y live in the same locality and y and z live in the same locality.


x and z live in the same locality


(x,z) ϵ R


R is transitive.


Therefore, R is reflexive, symmetric and transitive.


(c) It is given that R = {(x, y) : x is exactly 7 cm taller than y}


(x,x) R as human being x cannot be taller than himself.


R is not reflexive.


Now, if (x,y) ϵ R, then x is exactly 7 cm taller than y.


But y is not taller than x.


(y,x) R


R is not symmetric.


Now, let (x,y), (y,z) ϵ R


x is exactly 7 cm taller than y and y is exactly 7 cm taller than z.


x is exactly 14 cm taller than z.


(x,z) R


R is not transitive.


Therefore, R is neither reflexive, nor symmetric, nor transitive.


(d) It is given that R = {(x, y) : x is wife of y}


(x,x) R as x cannot be the wife of herself.


R is not reflexive.


Now, if (x,y) ϵ R, then x is the wife of y.


But y is not wife of x.


(y,x) R


R is not symmetric.


Now, let (x,y), (y,z) ϵ R


x is the wife of y and y is the wife of z.


This cannot be possible.


(x,z) R


R is not transitive.


Therefore, R is neither reflexive, nor symmetric, nor transitive.


(e) It is given that R = {(x, y) : x is father of y}


(x,x) R as x cannot be the father of himself.


R is not reflexive.


Now, if (x,y) ϵ R, then x is the father of y.


But y is not father of x.


(y,x) R


R is not symmetric.


Now, let (x,y), (y,z) ϵ R


x is the father of y and y is the father of z.


x is not the father of z.


Indeed x is the grandfather of z.


(x,z) R


R is not transitive.


Therefore, R is neither reflexive, nor symmetric, nor transitive.


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
Different kind of mappingsDifferent kind of mappingsDifferent kind of mappings58 mins
Functions - 06Functions - 06Functions - 0648 mins
Functions - 11Functions - 11Functions - 1156 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
Some standard real functionsSome standard real functionsSome standard real functions61 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