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I have question about set theory again. I have non-empty set $A$, and set $K$ of all equivalence relations on $A$. $K$ is partially ordered set regarding subsets ($\subseteq$). now I need to find the greatest element and the least element in $K$. so I think that the greatest element is the relation $A \times A$ and the least element is $\varnothing$, am I wrong?.

Now, they say that $A = \{1,2,3,4\}$. and they move the greatest element and least element to new set named $L$. the set $L$ is also partially ordered set regarding subsets ($\subseteq$). and I need to find two different maximal elements and two different minimal elements.

So if I understand correctly the question, the set $L$ should be $\{\varnothing, A \times A\}$. but how I can find two different elements in this set if the set consists of two elements?

I think there is something here that I don't understand. I will be glad if someone will tell me how to continue from here.

Thanks in advance.

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@user30648: (1) No, $\varnothing$ is not the minimal element: it's not in the set (Hint: Is $\varnothing$ a reflexive relation on $A$?) I think the set $L$ is supposed to be the set you obtain from $K$ by removing $A\times A$ and the minimal element of $K$. – Arturo Magidin May 4 '12 at 20:04

1 Answer 1

$\varnothing$ is not in $K$, unless $A=\varnothing$. Ask yourself this: if $\varnothing$ a reflexive relation on $A$ if $A\neq\varnothing$? So you'll need to think a bit more about it.

The second part is probably actually asking you to consider the set $L$ you obtain by removing the maximum and the minimum elements of $K$, rather than the set that contains only the maximum and minimum elements of $K$.

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