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# DM-Groups and Lattice-Q7

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For a positive number c, consider the open interval $G = (-c, c)$ of real numbers. For $x, y \in G$, define

$x*y:=\frac{x+y}{1+xy/c^2}$

Which one of the following statements is true?

(A).G is monoid but not group

(B).G is a group but not abelian

(C).G is semigroup but not a monoid

(D).G is an abelian group

reshown Jul 9, 2019

D is correct one
answered Jul 9, 2019 by (34,820 points)
what will be the identity element?
Zero
because smallest positive number is always greater than zero.
can you show how u did ? that would be very helpful
solution anyone???...

Let, $x, y \in G$
$\therefore x * y = \frac {x + y}{1 + \frac {xy}{c^2}} = \frac {y + x}{1 + \frac {yx}{c^2}} = y * x$

$\Rightarrow x * y = y * x$

Therefore, $G$ is an Abelian Group.

Also, the identity element in $G$ is $0$; and the inverse of an element $x \in G$ is $-x$.

answered Sep 15, 2019 by (210 points)