Q. 25 B5.0( 3 Votes )

# Let A = Q × Q and

Answer :

(i) Let (e, f) be the identity element for *

for (a, b) Q × Q

We have-

(a, b) * (e, f) = (a, b) = (e, f) * (a, b)

(ae, af+b) = (a, b) = (ea, eb+f)

From 1st & 2nd part-

ae = a e = 1 ...(1)

and, af+b = b af = 0 ...(2)

From 2nd & 3rd part-

a = ea e = 1 ...(3)

and, b = ef+b

b = (1)f + b [using (3)]

b = f + b

f = 0 ...(4)

From (2) & (4), we find that

a need not to be '0'.

e = 1, f = 0

(e, f) = (1, 0) Q × Q

(1, 0) is the identity element of A.

(ii) Let (a, b) Q × Q

Let (c, d) Q × Q

such that

(a, b) * (c, d) = (1, 0) = (c, d) * (a, b)

(ac, ad + b) = (1, 0) = (ca, cb + d)

ac = 1, ad + b = 0, ca = 1, cb + d = 0

c = (1/a), d = (-b/a), (1/a)b + d = 0 (a ≠ 0)

(c, d) = (1/a, -b/a) (a ≠ 0)

for a ≠ 0, (a, b)-1 = (1/a, -b/a)

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 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