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Let $A=\{a,b,c,d,e,f\}$, $R$ is defined by: $\{(a,b), (c,d),(d,c),(d,a),(d,b),(d,f),(e,a),(e,b),(e,c),(e,d),(e,f),(f,a),(f,b)\} \cup \Delta_A$.

I am not sure on how to draw a Hasse Diagram for $R$.

I start of with the non empty set which is in relation with $a,b,c,d,e,f$. Then the other ordered pairs would be $(a,b),(a,c),(a,d),(a,e),(a,f),(b,c),(b,d),(b,e),(b,f), (c,d),(c,e),(c,f),(d,e),(d,f),(e,f)$.

Is there another layer above these ordered pair. I don't knw how to complete the Hasse Diagram. Any help/information will help..

thank you!

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1 Answer 1

In the Hasse diagram (only applicable for partial orders), you only draw the covering relations, i.e. those relations where $x < y$ and there does not exist $z$ such that $x < z < y$. (Here I'm writing $x < y$ instead of $(x,y) \in R$.)

For example, you have that $a < b$ and $b < c$. These two relations imply $a < c$ when $R$ is a partial order. Thus $a < c$ is not a covering relation, and so you do not draw an edge connecting $a$ to $c$. You have to analyze further to see whether $a < b$ and $b < c$ are themselves covering relations or can be decomposed further. In this example, they are covering relations (why?), so you would draw an edge from $a$ to $b$ and from $b$ to $c$. Looking at those two edges in the Hasse diagram would then tell you that $a < c$.

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