Let $S = \{1,2,3\}$ and let the poset $(\wp(S)\setminus\{\emptyset\}, \sim)$ be defined as follows:
$$\begin{aligned} X \sim Y \Leftrightarrow X = Y \text{ or } \max(X) < \max(Y) \end{aligned}$$
Determine if $\sim$ is a total order relation and if $(\wp(S)\setminus\{\emptyset\}, \sim)$ is a lattice.
It's easy to show that $\sim$ is not a total order relation because if we consider $\{3\},\{1,3\} \in \wp(S)\setminus\{\emptyset\}$ then
$$\begin{aligned} \{3\} \neq \{1,3\} \text { and } 3 \nless 3 \Rightarrow \{3\} \nsim \{1,3\} \end{aligned}$$ $$\begin{aligned} \{1,3\} \neq \{3\} \text { and } 3 \nless 3 \Rightarrow \{1,3\} \nsim \{3\} \end{aligned}$$
What I really struggle with is showing if this poset is a lattice. By very definition, we know $(\wp(S)\setminus\{\emptyset\}, \sim)$ is a lattice $\Leftrightarrow \forall X,Y \in \wp(S)\setminus\{\emptyset\}$, $X \vee Y = \sup\{X,Y\}$ and $X \wedge Y = \inf\{X,Y\}$, but I can't figure out what $X \vee Y$ and $X \wedge Y$ pratically mean. Do I need to substitute $\sim$ for $\vee$ and $\wedge$?