Partial order

A relation $R$ on a set $A$ is called an partial order if it is: ?

• reflexive: everything loops
• antisymmetric: we can only travel one way
• transitive: every 2-step journey must be also done in 1-step

For example:

• $\le$, $\ge$, and ‘divisibility’ on ${\mathbb{N}}^{+}$
• Prove that $\subseteq$ is a partial order on the power set $P(S)$ of a set $S$.
  • $\subseteq$ is reflexive because $A\subseteq A$ for every set $A$.
  • $\subseteq$ is antisymmetric because if $A\subseteq B$ and $B\subseteq A$, then $A=B$ holds.
  • $\subseteq$ is transitive because if $A\subseteq B$ and $B\subseteq C$, then $A\subseteq C$ holds.