Question

What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set inclusion. Is R a partial order

Answer 1

$P(S)=\{\empty,\{x\},\{y\},\{z\},\{x,y\},\{x,z\},\{y,z\},\{x,y,z\}\}$

Define a relation R by A R B iff $A\sube B \ \forall\ A,B\isin P(S)$

We have $A\sube A$ for any $A\isin P(S)$

$\implies \sube$ is reflexive on $P(S)$

For any $A,B\isin P(S),\ A\sube B,B\sube A$ then $A=B$

$\implies R$ is anti symmetric on $P(S)$

For any $A,B,C\isin P(S),\ A\sube B,B\sube C\implies A\sube C$ then $A=B$

$\implies R$ is transitive on $P(S)$

So, $R(\sube)$ is a partial order on $P(S)$.