I'm currently studying for my exams and want to know if my proof of the following task is done correctly:
Prove, that the relation $R(G,\preccurlyeq_G) \; (G_1 \preccurlyeq_G G_2 \Leftrightarrow G_1 \text{is a subgraph of } G_2)$ of two graphs (unsigned, labeled) is a partial ordered set (Poset).
Poset$\implies$ Relation must be reflexive, antisymmetric and transitive.
Reflexive:
Let $G=(V,E): V=\text{Vertices},\quad E= \text{Edges}$
$(G_1,G_1)\Longleftrightarrow \text{Every graph is a subgraph of itself.}$
That means, that $(G_1\preccurlyeq_G G_1)\Longleftrightarrow V_1$ is a subset of $V_2 $ and $ E_1$ is a subset of $ E_2$
Antisymmetric:
$(G_1 \preccurlyeq_G G_2)$ and $(G_2 \preccurlyeq_G G_1) \implies G_1 = G_2$
$\Longleftrightarrow (V_1$ is a subset of $ V_2$ and $ E_1$ is a subset of $ E_2)$ and $(V_2$ is a subset of $ V_1$ and $ E_2$ is a subset of $ E_1)\implies G_1=G_2 \text{ because }V_1$ is a subset of $ V_2 $ and $ V_2$ is a subset of $ V_1 \implies V_1 = V_2$ (Analogous to $E$)
Transitive:
$(G_1 \preccurlyeq_G G_2)$ and $(G_2 \preccurlyeq_G G_3) \implies (G_1 \preccurlyeq_G G_3)$
$\Longleftrightarrow (V_1 $ is a subset of $ V_2 $ and $ E_1 $ is a subset of $ E_2) $and$ (V_2 $ is a subset of $ V_3 $ and $ E_2 $ is a subset of $ E_3) \implies (V_1 $ is a subset of $ V_3 $ and $ E_1 $ is a subset of $ E_3)$
If $V_1 $ is a subset of $ V_2$ and $E_1 $ is a subset of $ E_2$ would be the same as: $|V_1| \leq |V_2|$ and $ |E_1| \leq |E_2|$
Because $V_2 $ is a subset of $ V_3$ and $E_2 $ is a subset of $ E_3$ would be the same as: $|V_2| \leq |V_3|$ and $|E_2| \leq |E_3|$ is
$V_1 $ is a subset of $ V_3$ and $E_1 $ is a subset of $ E_3
\Longleftrightarrow |V_1| \leq |V_3| $ and $ |E_1| \leq |E_3|$
and that's why $(G_1 \preccurlyeq_G G_3).$
$\implies R(G,\preccurlyeq_G)$ is a $\textit{Poset}$
Kind regards,
Doesbaddel