Question

Punctul b) vă rog
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Answer 1

$\displaystyle\it\\x*y=\frac{xy}{2xy+1-x-y} \in G,~unde~G=(0,1),~pentru~oricare~ar~fi~x,y\in G.\\desigur,~ne~propunem~sa~aratam~ca~fractia~este~subunitara.\\pentru~asta~ne~folosim~de~punctul~a)\\x*y=\frac{xy}{xy+(1-x)(1-y)} \in (0,1) \Leftrightarrow \\ xy<xy+(1-x)(1-y) ~ |-xy \implies 0<(1-x)(1-y),\\inegalitate~evident~adevarata~pentru~x,y\in(0,1).\\~aratam~ca~x*y \neq 0~si~ca~x*y\neq 1.$

$\displaystyle\it\\daca~x*y\neq 0 \implies xy\neq 0,evident~adevarat~pentru~x,y\in(0,1).\\daca~x*y\neq 1 \implies xy\neq xy+(1-x)(1-y),~pp~prin~abs~\\ca~avem~egalitate.\\ xy = xy+(1-x)(1-y)~ |-xy \implies 0=(1-x)(1-y),~dar~x,y\in(0,1),~\\aceasta~egalitate~are~loc~daca~x~sau~y~era~1.\\prin~urmare~am~demonstrat~ca~x*y\in(0,1)~pentru~x,y\in(0,1).$