Find a relation over $P(${$1,2,3$}$)$ such that $|R|=12$ and the transitive closure of $R$ is the proper subset relation

Question

I'm having trouble finding relation $R$ over $P(${$1,2,3$}$)$ such that $|R|=12$ and the transitive closure of $R$ is $T$, the proper subset relation over $P(${$1,2,3$}$)$. My thoughts: a pair of subsets $(A,B)$ of {$1,2,3$} $\in$ $T$ if and only if $A \subset B$ and $A \ne B$.

Any hint would be useful.

Thanks.

Answer 1

Hint: Draw the lattice of subsets of $\{1,2,3\}$.

Answer 2

Hint. The number of true statements of the form $X\subset Y$, where $X,Y\in P(\{1,2,3\})$, is $19$. Write out all of these, then delete any that can be derived by transitivity from those you have not deleted.