Q. 93.8( 6 Votes )

Given a non-empty set X, consider the binary operation : P(X) × P(X) P(X) given by A B = A B A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation .

Answer :

It is given that : P(X) × P(X) P(X) given by

A B = A B A, B ϵ P(X).


As we know that,


A * X = A = x * A A ϵ P(X).


Thus, X is the identity element for the given binary operation *.


Now, an element A ϵ P(X) is invertible if there exists B ϵ P(X) such that


A * B = X = B * A (As X is the identity element)


A B = X = B A


This can be possible only when A = X = B.


Therefore, X is the only invertible element in P(X) w.r.t. given operation *.


Hence Proved.


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
Functions - 06Functions - 06Functions - 0648 mins
Functions - 11Functions - 11Functions - 1156 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