We have #G=(0,oo)#/#{1}# and #x@y=x^(lny)# .How you demonstrate that #forall x,yinG# then #x@yinG#?

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Answer 1 · 2017-05-27T14:52:37

The Proof is in the Explanation.

It is clear that, #AA x in (0,oo)-{1}, xoy=x^(lny)>0.#

What remains to be actually shown is #xoy!=1.#

Suppose, to the contrary, #xoy=1.#

#rArr x^lny=1.#

# rArr ln(x^lny)=ln1=0.#

# rArr (lny)(lnx)=0.#

# rArr lny=0. or, ln x=0.#

# rArr y=1, or, x=1.#

This contradicts the fact that, #x, y in (0,oo) -{1}.#

Hence, the Proof.