Show that inclusion relation ⊆ is a partial ordering on the power set of a set S. Draw the Hasse diagram for the partial ordering {(A,B) | A ⊆ B} on the power set P(S) where S = {a,b,c}.
Let us show that inclusion relation is a partial ordering on the power set of a set .
Since for any subset we conclude that this relation is reflexive.
Taking into account that and imply we conclude that the relation is antisymmetric. Since and imply , it follows that this relation is transitive.
Consequently, the inclusion relation is a partial ordering on the power set of a set .
Let us draw the Hasse diagram for the partial ordering on the power set where
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