Let $A,B\in\Bbb Q,\ \ A < B$ if $A\leq B\ \land \ A \neq B$
Let $\preceq$ be a ordering of a set $A$, and $B \subsetneqq A$.
$B$ is initial segment of $A$ under $\preceq $ if $(\forall a \in A, \forall b \in B)(a \preceq b \to a \in B )$
The following definitions are correct:
Let $\leq$ be an ordering of $\Bbb Q$, and $B \subsetneqq\Bbb Q$.
$B$ is Dedekind cut on $\mathbb{Q}$ if $B$ initial segment of $A$ under $\leq$ and $\forall C \in B, \exists D \in B (C < D)$
$\Bbb R:=\{B \subsetneqq \Bbb Q|B\text{ is Dedekind cuts on }\Bbb Q\}\ ?$
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Thanks in advance!!