 define common knowledge through this equation. An event A is said to be common knowledge in the state of the world omega star for all i1 dot dot dot i r say we have omega belongs to k i1 of k i2 dot dot dot k i r sorry omega star belongs to. So, what does this mean that it means that i1 knows that i2 knows that dot dot dot i r knows a in state of. So, in omega star they i1 knows that i2 knows that dot dot dot i r knows a and this is true for for all for all such sequences that you can create. So, this is the definition it will turn out that it reduces to something much simpler in certain cases. So, we will discuss that in subsequent. Let us take another extreme example. Let us take the example that let us take the event B. The event B is that the Paul is not colorblind and the purple car has one or let us say let us take a different color. Let us take Paul is not colorblind and the red car has one. So, now what when what does what does the knowledge operator say about Paul. So, when does Paul know this event? When does Paul know that he that he is not colorblind and the red car has one it is it is B itself right Paul will know. So, Kp of B is actually equal to B itself because when he is not colorblind and the red car has one he does know that he is not colorblind and the red car has one. What about John? So, when does John know that Paul is not colorblind and the red car has one empty set he will never know this right you can check this there is no element of the partition which which which contains which is completely contained in set B right. So, this is in fact empty. So, what this also means is therefore that this is also empty therefore that means there will never be a state of the world in which John will know that Paul knows that he is colorblind and the red car has one. So, there is no state of the world in which John will know that Paul knows that he Paul is not colorblind and the red car has one yes not necessarily yeah. So, this is just again an example yeah. So, this is this is an example this is a very simple case because the states are and the partitions are also kind of trivial in this case, but it does not this is not the only way in which it will happen it could very well happen for instance that this is that you know this here this here is not empty this one is also not empty, but this is empty. So, John will know something about about the event there will be some states of the world in which John will know the event there will be some states of the world in which Paul will know the event, but there will be no states of the world in which John will know that Paul knows the event is this clear that that could very well happen. Now, let us look at a few properties of the of the knowledge operator because this is such a important and useful tool. So, here is the first property you for any for any event and any player you always have k i of a is a subset of a for any event a and any player i k i of a is always a subset of a this means that. So, the left hand side here is the states of the world in which player i knows a. So, whenever player i knows a that state of the world is always a subset of a which means for player i to know a a should have obtained. So, a should have occurred or obtained. So, player i knows a implies a as a has actually occurred. So, what this effectively means is that the knowledge of we the players get knowledge of things when in out as a consequence of some physical reality. It is not what is knowable in principle, but what has actually happened and therefore they have come to know. So, they they then their knowledge the players player i knowledge of a is as has to be has to have is as a result of a occurring. So, therefore, so whenever player i knows a has actually occurred. So, this is this is therefore a different this framework as I had mentioned at the start that this is way more of a control theory type of model because you know of an event after it occurs. It is not about what could have been known or what could be knowable. So, that so control theory is always causal and that is how it this that is the way this model is also built. So, this is quick proof of this this this. So, omega is omega belongs to Ki of a which means basically that Fi of omega is a subset of a, but Fi of omega is this is a is the element of the partition that contains omega. So, which but but omega is always a is always in Fi of omega which means omega is in a. So, we start off from any omega in Ki of a, we get that omega must be in a. So, now let us suppose I gave you two events a comma b these are both sub events and a is a subset of b. So, a is a subset of b vertices mean whenever a occurs b also occurs. So, a is more specific than b, b is more general. So, whenever a has occurred b also occurs what this what would this what does the knowledge operator then say about knowing a and knowing b. So, whenever you know a you would also know b and this is. So, for any player Ki of a would always be a subset of Ki of b necessarily. Now, suppose so let us take any event. This is the event that player i knows a and this is the event that player i knows that player i knows a how are these related. So, player i knows a and player i knows that player i knows a what do these are superset. So, what is I mean it is really incredible you get this for free you get that this these two are equal these this is actually these are actually equal. So, our way of modeling I told you again that this is a kind of a physical way of modeling things because as a result of this self-awareness comes for free. Whenever player i knows an event he also knows that he knows that event. This is a consequence of the way we have modeled. This could have limitations but on the face of it for example loss of memory absent mindedness and so on cannot be will not satisfy something like this. So, if you have a device that has finite memory it may it may know something but may not know that it knows something and so on. But you will see later that this is general enough to model a whole lot of other things also. So, this is the self-awareness of a player you get for free from here. So, some more properties let us what is what is Ki of y it is y. So, every state of the what does this basically say in every state of the world player know every player knows that something has happened. So, what one consequence you get for free here is that y is common knowledge because Ki of y is y and this is true for every player. So, therefore every player knows that every player knows that every player knows that something has happened in every state of the world. Suppose a and b are two events then what is this player i knows that a has occurred player i also knows that b has occurred. So, when player i knows that a has occurred player i knows that b has occurred we can put two and two together and knows therefore that a and b have both occurred. So, we can therefore we are not proving this this is very easy to prove I am skipping the proof. So, this is all. So, the knowledge operator also satisfies this. First tell me I will just write this out you tell me if there is any what is the difference between these two what is the difference between Ki of a the whole complement and Ki of a complement. So, this is just again for clarity the left hand side is the set of states of the world in which player i does not know a. So, now the right hand side is the state is the states of the world in which player i knows that a has not occurred these are two very different things. So, therefore, there is no obvious relation between the two you can of course say one thing which is erase this this is in fact equal to is actually equal to this. So, when player does not know something he is not ignorance of ignorant of his own ignorance he also knows that he does not know. So, again as I said self-awareness sort of thing comes up the Ki of a is a Ki of a the whole complement is equal to Ki of Ki of a the whole complement. So, does not know is a slightly again. So, when we start discussing these subjects you know the English language is a little bit insufficient because what does it mean to not do when I say I do not know I do not know means I can for sure say that I cannot confirm that a has occurred this that is what this statement is saying. But the question he is asking is does the does Ki of a complement means that I does not know a is the states of the world in which I does not know a or is it the states of the world in which he may or may not have may may have not known a. So, the does not know so any partial knowledge in this model in this model there is no partial knowledge. If you know and if you do not if you if it is not true that you know then you do not know. Yes, but in that case he does not know that is what the that is our that is our definition. So, there is a more subtle model which will which we will do after this in which there is a probability that he assigns to each event that is what is called a belief. So, from knowledge you can move to beliefs when and which is in which a player basically assigns a probability distribution on the probability distribution on the states of the world and therefore on an event and so on. And there you can have these kind of graded sort of levels of knowledge. So, there knowledge becomes belief with probability one. Okay, but we will come to that after we have done with done with this. Yes, why is the last property true? We can we can show this. So, I will just clear out the proof here. So, if omega belongs to Ki of a no it does it is not sufficient the compliment is outside. So, of course, you would have this the claim is not about this. So, this let us just do this one second. So, so you have you have your a and then you have your fi of omega fi of omega is not a subset of a that means it it has something with some elements which are outside. Now, these are these elements which are outside are therefore elements of a compliment. So, Ki of a is a subset of a so Ki of a compliment is a superset of. So, I think it should follow from the first one I will just have to write it out properly. So, it is the Ki of a is a subset of a means this compliment is a subset is a superset of a compliment. And so, one of this Ki of a is a union of yes, yes Ki of a is all of these. So, in this case is a union of elements of a partition, correct, some other partitions that is right, correct, correct. So, I mean this I am I am just trying to see if it can be derived as a corollary of this. So, there is I am I am sure the result is true and in the reason for that is this only that you see whenever. So, this the knowledge the these the events when the should have picked a different color the states of the world in which player I knows a that means is that means Ki of a will necessarily be either all of fi of omega or none of that is how it will be you will never have this sort of space part in Ki of omega Ki of a for any for any event a because. So, all of see because the if this is included then this would also have to be included, but then this is not in a got it the nearby point which is in the same a partition same element of the partition but is not in a would also have to be included. So, only this part gets included and then, but I am trying to think is see if I can just apply one of these inclusion relations and derive this, but it seems like that is not possible. Anyway, we will come to we will do a much more complicated version of this subsequently in which this will come out for free actually we will write out the entire structure of this. So, I will do I will just remind me I will do this proof in class. So, actually let us do a quick property or another property of common knowledge. Let us come back to common knowledge and let us do one theorem related to it.