Question

Let \( \cdot \) be an associative operation on a non empty set A with identity e. Suppose a,b,c and d are elements of A, b is the inverse of a, and d is the inverse of c. Prove that db is the inverse of ac

Answer 1

OK, so (d.b).(a.c)=(d.c).(a.b)=e.e=e, so by definition d.b is an inverse of a.b

Answer 2

I mean a.c

Answer 3

simple as that

Answer 4

yayyyyyyyy

Answer 5

Thanks :))))

Answer 6

well i have a question

Answer 7

it just said db and you computed it as \( d \cdot b\)

Answer 8

Like i just checked my book and i saw a hint compute (ac)(db) and (db)(ac)

Answer 9

I used the dot to refer back to the fact that we are dealing with an operation. A binary operation is actually a relation that maps ordered pairs to elements. I don't know if that answers your question, I was looking at the wikipedia page for binary operation and I lost my train of thought. Good material, though:
http://en.wikipedia.org/wiki/Binary_operation

Answer 10

The book might have given such hint just because the defination says that if X and Y are inverse of each other, then X.Y=Y.X=Identity
The X.Y=Y.X part is just to show that the group also follows commutative property also.......but again using simple manipulation as the 1st,u can prove the 2nd part also....

Answer 11

Gotcha.Thanksss guyyss :)