I have a basic understanding of perfect secrecy. In the case where |K| == |M|, I see that there is only one key to encrypt a given m to a given c. Therefore each m is equally likely with same probability 1/|K|.
For the case of |K|<|M|, I have some intuition that it is not possible to map a random variable on set x to be a random variable on set y if |x| < |y|. But I'd like a formal proof of this. Or maybe this is the wrong way to look at this problem - I just want to see how we can prove that |K|<|M| result in non perfect secrecy.
This case of |K|<|M| is not a homework question; it is briefly mentioned in Boneh's Cryptography I, but not proved.