I have a question that I am stuck on reasoning about knowledge. For the following statement, I need to show whether it is valid, satisfiable or unsatisfiable:
$\bf{K}$(Happy $\lor$Sad) $\implies$ $\neg \bf{K}($Happy)
I have done the following:
Let e w be an interpretation where e is the set of possible worlds and w is the real world. Then we need to show that for any interpretation, e w satisfies $\neg \bf{K}$(Happy $\lor$Sad) $\lor$ $\neg \bf{K}$(Happy).:
Now assume that such an interpretation e w satisfies $\bf{K}$(Happy $\lor$Sad). This would mean for all w' $ \in$ e, e w' satisfies $\bf{K}$(Happy $\lor$Sad). However in a world w'' = {$\neg Happy, \neg Sad$}, then the interpretation e w'' does not satisfy $\bf{K}$(Happy $\lor$ Sad).
Therefore, we have a contradiction and for any interpretation e w, e w satisfies $\neg \bf{K}$(Happy $\lor$Sad).
Similarly, assume an interpretation e w satisfies $\bf{K}($Happy). However, if we have a world(s) where $\neg$Happy is true, then e w'' does not satisfy $\bf{K}($Happy). Therefore for any e w, e w satisfies $\neg \bf{K}($Happy) and thus the statement is valid.
I'm not sure whether my proof is correct and if not, could give me a hint on how to go about with this question.