So yeah, I have a problem, which goes like this :
Let A = {1,2,3,4}
K is a set of all symmetric, non reflexive relations :
$K \subseteq \ P(A \times A)$.
Meaning that for all $R \subseteq \ (A \times A), R\in \ K, (x,y)\in \ R \Longrightarrow (y,x) \in \ R$ and $I_{A} \notin \ R$
$ \subseteq \ $ is a partially ordered set on K
Now the last question is the one that confuses me and goes like this :
3)Prove that K does NOT have a Greatest Element( or Maximum Element).
Now I'm not sure why they're asking me to prove that there is no Maximum Element, since a relation that is Union of all relations in K should be the greatest relation which is a superset of all relations in K.
Example :
$ (R_{1},R_{2}) \in \ K$ symmetric and anti-reflexive.
$ K = \{R_{1},R_{2},R_{1} \cup R_{2}\} $
$ K_{\subseteq \ } = \{ (R_{1},R_{1}),(R_{1},\{R_{1} \cup R_{2}\}),(R_{2},\{R_{1} \cup R_{2}\}),(\{R_{1} \cup R_{2}\},\{R_{1} \cup R_{2}\})\} $
Note : Skipped empty set on purpose.
Clearly $R_{1} \cup R_{2}\ $ is the Greatest Element in K, since every element in K is a subset of $R_{1} \cup R_{2}\ $, and there's nothing else that $R_{1} \cup R_{2}\ $ can be a subset of, other than itself, which makes it the greatest element...
Am I missing something ?
Any help is appreciated!
Thanks.