Background
Consider a set of possible states, $W$. An agent, $i$ has knowledge of a fact $E\subseteq W$, $K_iE=\{w|\sim_i[w]\subseteq E\}$, where $w\in W$ and $\sim_i[w]=\{w'|(w, w')\in\sim_i\}$ is the set of all states indistinguishable from $w$ for agent $i$. (In other words, $i$ has knowledge of $E$ in state $w$ if $i$ also has knowledge of $E$ in state $w'$ for all $w'$ indistinguishable from $w$.)
A group of agents have general knowledge that $E$ at $w$ if all agents know $E$ at $w$. Mathematically, $$KE=\bigcap_i K_iE.$$
The problem
Consider the following situation: there are 3 states, $\{w_1, w_2, w_3\}\in W$, and two agents, Ann and Bob. For Ann, $w_1$ and $w_2$ are indistinguishable, so $\sim_{\text{Ann}}=(w_1, w_2)$. For Bob, $\sim_{\text{Bob}}=(w_2, w_3)$.
What is the general knowledge in state $w_3$?
My approach
We need to determine $K_\text{A}E$ and $K_\text{B}E$ at state $w_3\in W$.
At $w_3$, there are no states indistinguishable from $w_3$ for Ann, apart from $w_3$ itself. So $K_AE=\{w_3\}$.
For Bob, $w_2$ and $w_3$ are indistinguishable, so in state $w_3$, $K_BE=\{w_2, w_3\}$.
Hence, we get $KE=\{w_3\}\cap\{w_2, w_3\}=\{w_3\}$.
But on my lecture slides it says that the correct answer is $\{w_2, w_3\}$.
Questions
Where is my lack in understanding of how to correctly apply these mathematical definitions?
What is a good solution to obtain the correct answer?
Any general pointers in using these definitions would be useful for others too.
Edits/important points
$\sim_i$ is an equivalence relation and $\sim_i\subseteq W\times W$.
$KE$ is general knowledge, $K_i E$ is the knowledge of any single agent.