By definition $X\lor Y=\sup\{X,Y\}$ and $X\land Y=\inf\{X,Y\}$ if such elements exist, so you have to determine whether $\sup\{X,Y\}$ and $\inf\{X,Y\}$ exist for all $X,Y\in\wp(S)\setminus\{\varnothing\}$. Take $X=\{3\}$ and $Y=\{1,3\}$, for example. Is there a $Z\in\wp(S)\setminus\{\varnothing\}$ such that $X\sim Z$, $Y\sim Z$? If not, $X$ and $Y$ don’t even have an upper bound in $\wp(S)\setminus\{\varnothing\}$, let alone a least upper bound. If you can find an upper bound $Z$, you still have to show that $Z\sim U$ whenever $U$ is an upper bound for $X$ and $Y$, i.e., whenever $X\sim U$ and $Y\sim U$.
Added: Take the example of $X=\{3\}$ and $Y=\{1,3\}$. $Z$ is an upper bound for $X$ and $Y$ if $X\sim Z$ and $Y\sim Z$. You already know that $X\not\sim Y$, so $Y$ is not an upper bound for $X$ and $Y$. Similarly, $Y\not\sim X$, so $X$ isn’t an upper bound for $X$ and $Y$. Suppose that there is some $Z$ that is an upper bound for $X$ and $Y$; we’ve just seen that it can’t be $X$, so the only way to have $X\sim Z$ is to have $\max(X)<\max(Z)$. Similarly, we have to have $\max(Y)<\max(Z)$. But $Z\subseteq\{1,2,3\}$, so $\max(Z)\le3=\max(X)=\max(Y)$: it’s impossible to find a $Z\in\wp(S)\setminus\{\varnothing\}$ such that $\max(X)<\max(Z)$. We’ve now shown that $X$ and $Y$ have no upper bound in $\wp(S)\setminus\{\varnothing\}$. Since they have no upper bound at all, they certainly have no least upper bound: $\sup\{X,Y\}$ doesn’t exist. That’s just another way of saying that $X\lor Y$ doesn’t exist. And this means that the poset is not a lattice.
Note that some pairs do have least upper bounds. For instance, let $X=\{1\}$ and $Y=\{1,2\}$. $X\sim Y$ and $Y\sim Y$, so $Y$ is an upper bound for $X$ and $Y$. Moreover, if $Z$ is any upper bound for $X$ and $Y$, then $Y\sim Z$, so $Y$ is the least upper bound for $X$ and $Y$: $\sup\{X,Y\}=Y$.
Here are three little exercises for you to try on the basis of the ideas above; I’ll be happy to answer questions about them.
Do $X=\{2\}$ and $Y=\{2,3\}$ have a greatest lower bound? If so, what is $\inf\{X,Y\}$?
What about $X=\{1\}$ and $Y=\{1,3\}?$
Let $X=\{1\}$ and $Y=\{2\}$; what is $\sup\{X,Y\}$, if it exists? What about $\inf\{X,Y\}$?
Best Answer
Assuming you're using the poset definition of lattice . . .
Let $c = a \land (a \lor b)$.
The goal is to show $c=a$.
From $c = a \land (a \lor b)$, it's immediate that $c \le a$.
But also, $a \le a$, and $a \le a\lor b$, hence $a \le c$.
Therefore $c = a$.
An analogous argument proves the other absorption law.
You try it.