Q. 55.0( 1 Vote )

Let ‘o’ be a bina

Answer :

We are given with the set Q0 which is the set of non - zero rational numbers.


A general binary operation is nothing but association of any pair of elements a, b from an arbitrary set X to another element of X. This gives rise to a general definition as follows:


i. A binary operation * on a set is a function*:


Here the function o:


For the ‘o’ to be commutative, aob = boa must be true for all a, b Q0. Let’s check.



a * b = b * a (as shown by 1 and 2)


Hence ‘o’ is commutative on Q0


For the ‘o’ to be associative, ao(boc) = (aob)oc must hold for every a, b, c Q0.







Hence ‘o’ is associative on Q0


ii. Identity Element: Given a binary operation*: A X A A, an element e A, if it exists, is called an identity of the operation*, if a*e = a = e*a a A.


Let e be the identity element of Q0.


Therefore, aoe = a (a Q_0)








iii. Given a binary operation with the identity element e in A, an element a A is said to be invertible with respect to the operation, if there exists an element b in A such that a * b = e = b * a and b is called the inverse of a and is denoted by a–1.


Let us proceed with the solution.


Let b Q0 be the invertible elements in Q0 of a, where a Q0.


a * b = e (We know the identity element from previous)






Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
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
RELATED QUESTIONS :

| Let * be a binaMathematics - Board Papers

Find the idMathematics - Board Papers

Let f : A Mathematics - Exemplar

Show that the binMathematics - Board Papers

Determine whetherRD Sharma - Volume 1

Fill in theMathematics - Exemplar