wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Let A=Q×Q, where Q is the set of all rational numbers, and * be a binary operation on A defined by (a,b)(c,d)=(ac,b+ad) for (a,b),(c,d) ϵ A. Then find
(i) The identify element of in A.
(ii) Invertible elements of A, and hence write the inverse of elements (5,3) and (14,4).

Open in App
Solution

(a,b)(c,d)=(ac,ad+b)
(1) identity element
let (e,f) be the identity element .
Then (a,b)(e,f)=(ae,af+b)=(a,b)
[ae=a, where e is the identity element ]
(ae,af+b)=(a,b)
ae=a and af+b=b
e=1 and f=0
(1,0) is the identity element .
(2) invertible element
Let (c,d) be the inverse of (a,b)
then
(a,b)(c,d)=(1,0)[ab, e is the identity ]
(ac,ad+b)=(1,0)
ac=1;ad+b=0
c=1a,d=ba
So, (1a,ba) is the inverse.
The inverse of (5,3) is (15,35)
The inverse of (14,4) is (4,16)

flag
Suggest Corrections
thumbs-up
0
similar_icon
Similar questions
View More
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Operations on Sets
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon