Answer to Question #263006 in Discrete Mathematics for gikovi

Question #263006

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}.


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Expert's answer
2021-11-09T11:42:29-0500

Let us show that inclusion relation is a partial ordering on the power set of a set SS.

Since AAA\subseteq A for any subset AS,A\subseteq S, we conclude that this relation is reflexive.

Taking into account that ABA\subseteq B and BAB\subseteq A imply A=B,A=B, we conclude that the relation is antisymmetric. Since ABA\subseteq B and BCB\subseteq C imply ACA\subseteq C , it follows that this relation is transitive.

Consequently, the inclusion relation is a partial ordering on the power set of a set SS.


Let us draw the Hasse diagram for the partial ordering {(A,B)AB}\{(A,B) | A ⊆ B\} on the power set P(S)P(S) where S={a,b,c}.S = \{a,b,c\}.





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