Consider the set $ X = \mathcal P ( \{ 1 , 2 , 3 , 4 \} ) $ and the binary relation $ S $ on $ X $ such that $ A \mathrel S B $ is defined as $ A \cap \{ 1 , 2 \} \subset B \cap \{ 1 , 2 \} $.
I need to prove that $ S $ is a partial order and find minimum and maximum elements if they exist.
I showed anti-reflexivity and transitivity, but struggle to find min and max elements.
I thought maybe this could be the max sets but I may be wrong
$$\{1,2\},\{1,2,3\},\{1,2,4\},\{1,2,3,4\} $$