Show x+y/xy is not a group

Question

οHi,

I need help to show that this is not a group

(positive real numbers, ο) where xοy=(x+y)/xy

I know it's closed as x and y are positive real numbers, so not equal to 0, so xy cannot be 0, therefore (x+y)/xy is also in the set of positive real numbers.

It's with the identity, inverse and associativity that i am struggling.

Any help would be appreciated.

Answer 1

It follows if you observe that there is no element x in the positive real numbers that satisfies the equality xo1=1. To see this notice that xo1=(1/x)+1 and if this equals 1 then we must have that there is a positive real number x such that 1/x=0. This is wrong of course and hence, the set of positive real numbers with the operator o is not a group.