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.