Q. 84.2( 72 Votes )

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Answer :

It is given that the relation R in the set A = {1, 2, 3, 4, 5}

given by R = {(a, b) : |a – b| is even},


It is clear that for any element a ϵ A, we have |a – a|= 0(which is even)


Therefore, R is reflexive.


Now, Let (a,b) ϵ R


|a b| is even


|-(a b)| = |b - a| is also even.


(b,a) ϵ R


Therefore, R is symmetric.


Now, Let (a,b) ϵ R and (b,c) ϵ R .


|a b| is even and |b c| is even


(a – b) is even and (b – c) is even


(a c) = (a b) + (b c) is even


|a c| is even


(a,c) ϵ R


Therefore, R is transitive.


Therefore, R is an equivalence relation.


Now, all elements of the set {1,3,5} are related to each other as all the elements of this subset are odd.


Therefore, the modules of the difference between any two elements will be even.


Similarly, all elements of the sets {2,4} are related each other as all the elements of this subset are even.


Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2,4} are even.


Therefore, the modulus of the difference between the two elements will not be even.


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
Range of FunctionsRange of FunctionsRange of Functions58 mins
Quick Revision of Types of RelationsQuick Revision of Types of RelationsQuick Revision of Types of Relations59 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
Understand Fractional part and Mod Function in 1 hourUnderstand Fractional part and Mod Function in 1 hourUnderstand Fractional part and Mod Function in 1 hour61 mins
Battle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their Graphs48 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