$.$ Consider the set $\{$x in R : $0<=$ x $<1\}$ and the operation x $$ y $=$ x $+$ y $($x $+$ y$).$ Show this is a group.

The operation is associative, but it$$s a pain to show it$,$ so just show the other three properties. Note

that $$x$$ denotes the $$floor$$ of x$,$ gotten by dropping its decimal part. For instance, $1\mathrm{.}73=1.$