Your solution is not right in that in the first paragraph you generalise, not showing a particular M and immediately in the second paragraph you give specific values for M.y does not satisfy b and also the part in your second paragraph where you say "since y is accessible from x; ¬(a -> b) fails to be satisfied at y." is not right.
What you should take note of is as follows:
Given some sentence
S of a Modal Language KL
ALL M= (G, R, V) and this is allowed.
There are only 2 possibilities:
1. S is globally true in
all M (which means in every world w of G for every M). To prove this (when this is the case) you say
Let M= (G, R, V) be any modal model (which means we do not assume anything about M) and w be any element of G. We show that S is true in w.... you can give an example.
2. The other possibility is that S is not globally true i.e. there exist an M= (G, R, V) and a w which is an element of G such that S is false in w. To prove this, you give M= (G, R, V) and w explicitly and prove that S is false in w.
There are a variety of examples given in your study guide on page 276-277, which gives you a clear picture. Study them.
Lecturer COS 361-F