 Hello and welcome to the session. In this session, first we will discuss de Moulton's laws. Consider a Boolean algebra consisting of the set B and binary operations of sum and product. Now consider two elements x and y belonging to the set B. Now from the de Moulton's laws, we have x plus y the whole complement is equal to x complement into y complement and x into y the whole complement is equal to x complement plus y complement. So these are the de Moulton's laws. Let us now see the proof of these laws. First of all, let us try to prove x plus y the whole complement is equal to x complement into y complement. This means we have to show that x complement into y complement is the Boolean inverse of x plus y. So in order to prove this, we will prove that x plus y plus x complement into y complement is equal to 1 plus y the whole into x complement into y complement the whole is equal to 0. If we prove these two, then this would imply that x plus y the whole complement is equal to x complement into y complement. So now first of all, we consider x plus y the whole into x complement into y complement the whole. Now this would be equal to x complement into y complement the whole into x plus y the whole. The operation of product is commutative. So using the commutative law here, now consider this distributive law which shows the distributivity of the product over the sum. So using this distributive law here we get this is equal to x complement into y complement the whole into x plus x complement into y complement the whole into y. This is by using the distributive law. Now as the operation of the product is commutative, so applying the commutative law here we have y complement into x complement the whole into x plus x complement into y complement the whole into y. This is by using the commutative law. Now this is the associative law in which we have a into b the whole into c is equal to a into b into c the whole. Now using this associative law here we get this equal to y complement into x complement into x the whole plus applying the associative law for this term also we get x complement into y complement into y the whole. So this is by using the associative law. Now by the property of inverse we have a into a complement is equal to 0 which is same as a complement into a. So using this property of inverse for this that is x complement into x and y complement into y we have this is for the equal to y complement into x complement into x is 0 plus x complement into y complement into y is 0. So here we have used the property of inverse. Now further this is equal to y complement into 0 would be 0 plus x complement into 0 would be 0 by using the property a into 0 is equal to 0. And further this would be equal to 0 and this is the identity element for the operation of the sum. Therefore we now have x plus y the whole into x complement into y complement the whole is equal to 0 which is the identity element for the operation of sum. So we have proved this second result. Now let's prove this first one for this consider x plus y the whole plus x complement into y complement the whole. Now this is the distributive law which shows the distributivity of the sum over the product. So using this distributive law we have this is equal to x plus y plus x complement the whole into x plus y plus y complement the whole. This is using the distributive law. The sum is commutative so we have y plus x plus x complement the whole into x plus y plus y complement the whole. So this is using the commutative law for this that is x plus y becomes y plus x using this law and so further by the property of inverse we have a plus a complement is equal to 1 which is same as a complement plus a. So using this property of inverse we have y plus 1 that is x plus x complement would be 1 this whole into x plus 1 the whole that is y plus y complement would be 1. So this is using the property of inverse further we have y plus 1 would be 1 into x plus 1 would be 1. So we have 1 into 1 this is using the property a plus 1 equal to 1. Further 1 into 1 is equal to 1 which is the identity element for the operation of the product. Therefore we have shown x plus y the whole plus x complement into y complement the whole is equal to 1. So we have shown this also. Now since we have shown x plus y the whole plus x complement into y complement the whole equal to 1 and x plus y the whole into x complement into y complement the whole is equal to 0. Thus we can now say that x complement into y complement is the Boolean inverse of x plus y and therefore we now have x plus y the whole complement is equal to x complement into y complement and this is what we were supposed to prove for this De Morgan's law. Now next we need to prove that x into y the whole complement is equal to x complement plus y complement. So in the second part we need to prove x into y the whole complement is equal to x complement plus y complement. Now this result that is x into y the whole complement equal to x complement plus y complement is the rule of the first part that is x plus y the whole complement equal to x complement into y complement. Now in this result if we replace plus y the dot and dot by the plus we would get the second part. Hence De Morgan's laws. Next we discussed uniqueness of complement. Here we have a Boolean theorem according to which we have in a Boolean algebra the inverse of each element is unique. Let us see the proof of this now. We had considered a Boolean algebra which said to be and binary operations of sum and product. Now we consider x to be any element belonging to the set B. Now we suppose x1 and x2 are both inverses of the element x of set B. Now as x1 is inverse of x so this means x plus x1 is equal to 1. Now as x2 is inverse of x so x plus x2 is equal to 1 also x into x1 would be equal to 0 and x into x2 would be equal to 0. Now consider x1 this could be written as x1 into 1 this is by the property of identity. Now further in place of this 1 we can write x plus x2 so we have x1 into x plus x to the whole is equal to x1. Further we use this distributive law showing the distributivity of product over the sum that is this law we get x1 equal to x1 into x to the whole plus x1 into x to the whole. This is using the distributive law further we get x1 equal to x into x1 the whole plus x1 into x2 here we have used the commutative law. Now further we have x into x1 is 0 so here we have x1 equal to 0 plus x1 into x2 also we have x into x2 is equal to 0 so further x1 equal to x into x2 plus x1 into x2 next we have x1 equal to x2 into x plus x2 into x1 this is by the commutative law as the product is commutative so we would get this. Now we would use this distributive law to the right hand side so we have x1 equal to x2 into x plus x1 the whole this is using the distributive law. Here x plus x1 is 1 so further we have x1 equal to x2 into 1 so now we can say x1 is equal to x2 as x2 into 1 would be x2 by using the property of identity so we have x1 equal to x2 and we had supposed x1 and x2 to be inverses of the element x of the set B and now they are equal which shows that every element in the Boolean algebra has a unique inverse or say complement so this completes the session hope you have understood the D Morgan's law and the uniqueness of the complement or the inverse.