 lattices to a special class of lattices called Boolean algebra. Now as we have already seen that a lattice is a partially ordered set written as poset with the property that any two elements in it has a greatest lower bound and the least upper bound. We can rephrase whatever we have said now as suppose L is a lattice with a partial order denoted by the symbol less or equal then for any two elements a b inside L we can define an element called the greatest lower bound of a b in short glb of a b greatest lower bound of a b which we are writing as glb of a b and least upper bound of a b we can as well write it as a set a b that is lub of a b. Now what the definition of a lattice tells us is that in a lattice any two elements will have a greatest lower bound and the least upper bound because of this we can in fact write this glb and lub as binary operations over L we will denote them by symbols like v and wedge. So glb of a b is denoted by a and a wedge then b we will also sometimes denote it just by a dot b and lub that is the least upper bound of a b is denoted by a v v we sometimes will be denoting by a plus b now this a dot b or this a wedge b is called conjunction conjunction or the meet operation and a vb or a plus b is called disjunction or the join operation. Now in this lecture we will be using plus and dot instead of the v and wedge now there are certain properties of this meet and join that we will discuss now one is idempotent or idempotent c property this states that a plus a equal to a and a dot a is equal to a now this does not come as a surprise to us because we know that when we take the plus operation it is essentially the least upper bound of the elements now if we have if my set is single term so I have got a and a so the least upper bound is a itself and similarly for greatest lower bound the next property that we discuss is commutativity if we have two elements a and b inside l then a plus b is equal to b plus a and a dot b equal to b dot a this is also somewhat easy to think of because a plus b is nothing but the least upper bound of the set a b and as we know the elements of a set are not ordered so least upper bound of a b is same as the least upper bound of b a similarly a dot b is the greatest lower bound of the set a b which is same as the greatest lower bound of the set b a so these two the first two properties of this glb and lub operations need no proof as such now we come to another property which is called absorption now this absorption property needs a proof well we are starting with two elements a and b and we see that a dot b is a definitely something which is less or equal a the reason is that a dot b is the greatest lower bound of a and b and of course a will dominate a dot b and therefore if we consider the least upper bound of the set a and a dot b right so if we consider the least upper bound lub this is nothing but a plus a dot b this of course will give me a therefore I get the first absorption rule and then in the next in the next case we note that a plus b is definitely dominating a the reason is that if you consider a plus b it is the least upper bound of a and b therefore a will be less or equal to a plus b but then if we compare a and a plus b then the greatest lower bound of the set a and a plus b is going to be a in other words a dot a plus b is going to be a so these are the first two absorption laws over here and next we move on to associativity which states that a plus b plus c is equal to a plus b plus c and a dot b dot c is equal to a dot b dot c now in a lattice even these rules need a proof I will give the proof of the first associativity law and then leave the other part for exercise now let us look at the rule again we want to show that a plus b plus c equal to a plus b plus c in order to do that we first start by checking a a is definitely less or equal to a plus b and a plus b is definitely less or equal to a plus b plus c well the reason is that a plus b is the least upper bound of a and b therefore it must dominate a and a plus b plus c that I write here is the least upper bound of a plus b and c therefore it must dominate a plus b so by using the transitive law now because we know that after all we are looking at partial order relations using transitivity of the partial order we have a less or equal a plus b plus c now let us look at b and c now let us put this first in a box and then start off with b now b is definitely less or equal a plus b and a plus b is less or equal a plus b plus c therefore we can write again by using transitivity of the partial order involved b less or equal a plus b plus c we again put it in a box the next candidate is c start with c and of course c is less or equal a plus b plus c we put it in a box now we would like to combine these two results if you look at this we will see that in the both two boxes the right hand side is same that is a plus b within bracket plus c and b is less than a plus b plus c c is also less or equal a plus b plus c now we know that we are working in a lattice therefore b and c will have a least upper bound definitely a plus b plus c is an upper bound for the set bc we write it down a plus b plus c is an upper bound for the set bc therefore any least upper bound will be dominated by this upper bound so we know that the existence of the least upper bound of b and c is guaranteed so which we denote by b plus c therefore b plus c will be less or equal a plus b plus c why the reason is that since b plus c is the least upper bound of the set containing bc now we put a bracket over here as well now we see this and this again we see that a plus b plus c dominates a that is a is related to a plus b plus c it is less or equal a plus b plus c whatever we may say remembering that it is a lattice and b plus c is also less or equal a plus b plus c now of course we can we are we can take the least upper bound of a and b plus c which is a plus b plus c and this will be less than a general upper bound any general upper bound and one of them is a plus b plus c thus we see that we have been able to prove that a plus b plus c is related to a plus b plus c or less or equal a plus b plus c as we choose to say in a lattice now what about the other way round now if we look at this last relation we will see that in the bracket in the left hand side we have bc and in the right hand side we have a b and outside we have a and c that is the remaining symbol so this gives an idea what about if we exchange a and c that is exchange a and c where let us call it equation 1 or it is not an equation it is called it the relation 1 in 1 if we do that we will get c plus b plus a is less or equal c plus b plus a rearranging the terms using commutativity and yeah only we need only commutativity we will get a plus b plus c so I am using commutativity here and commutativity here is less or equal again using commutativity a plus b plus c so let us call it 2 so we have got now 2 and we have got 1 we remember that after all we are working in a lattice so therefore it is anti-symmetric so these 2 relations together imply that a plus b plus c is equal to a plus b plus c thus we have proved just now that in a lattice associativity holds for addition and what I leave as exercise is the meet operation or product whatever you say to prove that a dot b dot c equal to a dot b dot c what I can tell you is that it is the almost the same the underlying ideas are more or less same now now we make a statement that we do not prove explicitly but it is more or less clear we are considering finite lattices we state that in a finite lattice in a finite lattice there exists a unique least element and a unique greatest element the question is why the reason is that given any two elements I can find a least element and if I just pick up all the elements and mutually take least element and the least element of the least element ultimately I will arrive at the least so let me reward it given any two elements I get a least upper greatest lower bound so if I if I keep on doing that for all the all the elements and then construct all the greatest lower bounds and then do the same thing for all the greatest lower bounds so ultimately I will arrive at one element which is the least element in the whole lattice the reason is that we do not have infinitely many elements so whichever chain by using whichever chain we go downward we will stop at one element and which is going to work as a least element of the whole lattice and similarly we have the idea of least upper bound so we go on like that and get a greatest element this unique least element is denoted usually by the symbol 0 this unique least element is denoted by the symbol 0 and the greatest element by the symbol 1 and given any element a in the lattice L we have the properties a plus 0 is equal to a and a dot 1 is a dot 0 is 0 and a plus 1 is 1 these are the things which are again more or less straightforward because after all 0 is the greatest element so if I take the least upper bound of a and 0 it is sorry 0 is the least element so if I and so whatever element I take a it is going to dominate 0 therefore if I take the least upper bound it is going to be a and 1 is the greatest element therefore if I take the lower bound of 1 and a for any a it is going to be a similarly the other two cases now we come to the question of distributivity in a lattice now a lattice is called a distributive lattice if for any ABC in that lattice and I must mention here that ABC's need not be distinct we must have a plus b dot c equal to a plus b dot a plus c and a dot b plus c equal to a dot b plus a dot c now the next natural question that comes here is that is any lattice distributive and the answer is no there are lattices which are not distributive I will give examples of two lattices which are not distributive so first one is like this so suppose this is a this is b this is c this is well this is c so this is c this is d and this is e and I connect them and this gives me a has a diagram now let us consider the product b well I am using the other notation so I change my notation let us consider b plus c alright now let us evaluate c dot d the c and d this is a and b plus a gives me b so I can write that b plus c d is b if we consider b plus c and b plus d this will mean b plus c is e and b plus d is e therefore e meet e gives me e we put this in a box comparing these two we see that of course b is not e therefore in this case b plus c d is not equal to b plus c meet b plus d so this lattice is not distributive next we check another lattice which is not distributive which is given by a has a diagram like this we have now five elements we have got a over here then b then d and at the top well let me write e here because I will put c over here now let us see whether we can find out some combinations where which will not be a which will not be distributive so let us start from here d and then let us consider d plus b e now this is d plus this is not e this is c so this is c d plus b c now b c is a so d plus a so this gives me d on the other hand d plus b gives me d and d plus c gives me e and if I take a product d plus b product d plus c well this I see that this does not work out I get again d so it does not work out so we need some other combination let us let us try something else let us let us try the other way so let us consider d dot b plus c which is d meet b plus c is e this gives me d and d meet b plus d meet c d meet b will definitely give me b and d meet c gives me a so this gives me b alright so here we see that for this particular case this does not work out but if we change the plus with dot and dot with plus when we see that we have a situation here where of course b is not equal to d therefore we have d b plus c equal to db plus dc they are not equal thus we see that this lattice as well as the one we discussed previous to this both are non-distributive so these are examples of non-distributive lattices however for our purpose we will be considering only distributive lattices and not only that distributive lattices which has complements so we define another operation on a lattice which has got a greatest element and a least element so I am considering a lattice L with a partial order which is distributive has greatest and least elements denoted by 1 and 0 respectively we are considering this and we define unary operation on it which is called complement for any element a belonging to L the complement of a in L is an element a prime we will be writing this as a prime of the original element such that a plus a prime is equal to 1 and a dot a prime is equal to 0 if a lattice has a if a lattice is such that any element of it has a complement then it is called a complemented lattice we have to note here is that this complementation is not restricted to the distributive lattices we can have complemented lattices which are not distributive in fact a closer analysis will show us that the two nondistributive lattices that we constructed just before are complemented lattices but now we are not considering that we are considering distributive lattices having greatest and least elements and which are complemented which is a complemented lattice as well and these lattices are called Boolean algebras. So I am now in a position to define a Boolean algebra Boolean algebra is a complemented distributive lattice the operations plus dot and complementation defined as above here in all this context we must remember that we are not only looking at a set we have a set but on that set there are operations there are elements and all that and together it is forming an algebra. So suppose I have a set B on the set B we define a partial order suppose with respect to that partial order it becomes a lattice and because it becomes a lattice we have two operations defined on it that is plus and dot which is the least upper bound and greatest lower bound and then suppose the greatest element and least element exist and we can define a complement operation which is a unary operation and suppose it is a complemented lattice with respect to the this complementation and that complemented lattice is a distributive lattice we have discussed the distributive law. So we put all these operations together with the symbols for the greatest element and least element and what we get is called a Boolean algebra and we can denote it by a script B now the question is what is the simplest possible Boolean algebra and we indeed know probably the simplest possible algebraic structure that can ever be and that is a Boolean algebra that contains only two elements here please note that if we have just a set with one element we can have nothing so we have a set with two elements let us called them 0 and 1 call the set B and define plus as this 0 plus 0 is 0 0 plus 1 is 1 1 plus 0 is 1 and 1 plus 1 is 1 and let us define dot in this way 0 1 0 1 0 0 0 1 is 0 1 0 is 0 and 1 1 is 1 and let us define complement the unary operation in in this way that 0 is changed to 1 and 1 is changed to 0 and what we have is Boolean algebra and this is denoted by you know we can denote it by 0 1 which we are then plus dot this complement 0 1 so this all together give me gives me B now if we want to generate more Boolean algebras the best way is to take Cartesian product of this B so for example we can take B 2 which is the Cartesian product of B and B then we get this set well 0 0 0 1 1 0 1 1 plus dot complement and the least element is 0 0 greatest element is 1 1 now one can ask a question over here that what what do I mean by this plus and what do I mean by this dot and the complement operation here we note that if we consider just the set let us say B which is essentially 0 0 0 1 1 1 1 0 and 1 1 any two general elements of B can be written as x 1 x 2 and y 1 y 2 both belonging to B then x 1 comma x 2 y 1 comma y 2 belongs to I am sorry I change it this is B square this is B cross B this is B cross B so here it is B square but this x 1 x 2 y 1 y 2 are in B so x 1 x 2 will behave as elements in B therefore we can define the at the plus and dot by using the operations on B that is x 1 x 2 plus y 1 y 2 will be x 1 plus x 2 y 1 plus y 2 no this is this is not correct so I will change it this will be x 1 plus y 1 x 2 plus y 2 now we do not have to worry about this plus this plus is already defined over here and this is the plus that we are defining on B cross B by using the addition that we have already defined similarly we can define x 1 comma x 2 meet y 1 y 2 as x 1 y 1 dot here x 2 dot y 2 and we again remember that this meet or dot or product whatever we say is same as this one and we can check that the operations defined in this way follow the rules of Boolean algebra and for the complement again if we have x 1 x 2 complement we will just define it as x 1 complement x 2 complement thus we see that given our very small and simple Boolean algebra we can take on take Cartesian products of this and generate new Boolean algebras and we will see later on that any finite Boolean algebra is essentially Cartesian products of B with respect to the operations that I have just now described for B2 so for today we shall stop in the next lecture we will continue our discussions on Boolean algebras and functions on Boolean algebras thank you