\textbf{2.1. $\succ\cup\sim=\succeq$ }\&\textbf{ $\succ\cap\sim=\emptyset$}.

\textbf{2.2. $\sim\left(x\right)=\succeq\left(x\right)\cap\preceq\left(x\right)$}.

\textbf{2.3.} $\succeq\left(x\right)=\sim\left(x\right)\cup\succ\left(x\right)$.

\textbf{2.4.} Every relation $\succsim$ can be decomposed into a
symmetric $\sim$ and assymetric relation $\succ$ such that $\sim\cup\succ=\succsim$. 

\textbf{2.5.} Rational $\succsim$ implies transitive $\sim$. \\
\textbf{2.6.} Rational $\succsim$ implies transitive $\succ$.\\
\textbf{2.7.} Rational $\succsim$ implies that if $x\succ y\sim z$
then $x\succ z$.

\textbf{2.8.} A transitive, assymeteric relation has no cycles\emph{. }

\textbf{2.9.} If $\succsim$ is rational,\textbf{ }\emph{$\forall x,x'\in X$:$\succeq\left(x\right)\subseteq\succeq\left(x'\right)\lor\succeq\left(x'\right)\subseteq\succeq\left(x\right)$.}