Incomparable Elements In A Poset The problem I am working on is, Find two incomparable elements in these posets. a) $(P(\\{0,1,2\\}),⊆)$ b) $(\\{1,2,4,6,8\\},|)$ For a, I said that $R \subseteq p(\\{0,1,2,3\\}) \times p(\\{0,1,2,3\\})$, where $A$ and $B$ are sets, that are elements of the powerset. Then, $R=\\{(A,B)|A \subseteq B\\}$. An example of two incomparable elements would be $\\{0\\}$ and $\\{1\\}$, because they are not subsets of each other. So, the ordered pairs $(\\{0\\},\\{1\\})$ and $(\\{1\\},\\{0\\})$ are two ordered-pairs that contain elements incomparable to each other. (Is that proper to say that?) Would this be an acceptable answer? I don't like my textbook's solution: they never use any notation; there answer is completely descriptive, which is nice, but I would like if they supplemented the description with notation. I don't need help with part b, because if I answered part a correctly, then I will have answered part b correctly.

$\\{0\\}$ and $\\{1\\}$ are indeed incomparable elements of $\wp(\\{0,1,2\\})$ with respect to the partial order $\subseteq$, and for the reason that you gave: $\\{0\\}\
subseteq\\{1\\}$, and $\\{1\\}\
subseteq\\{0\\}$. There’s no reason to look at the ordered pairs, though it’s true that neither $\langle\\{0\\},\\{1\\}\rangle$ nor $\langle\\{1\\},\\{0\\}\rangle$ belongs to the order $\subseteq$.