 Hello and welcome to the session. In this session we discussed the following question with phase in the Boolean algebra consisting of the set B and operations sum, product and complement show that x plus x complement into y the whole into x complement plus x into y the whole is equal to y for all x and y belongs to the set B. Let us now proceed with the solution. Now we need to prove that x plus x complement into y the whole into x complement plus x into y the whole is equal to y where these x and y are the elements of the set B. For this we consider x plus x complement into y the whole into x complement plus x into y the whole. Now we have a distributed law according to which we have if we set B and the two binary operations sum and the product is a Boolean algebra then in this case we have a distributed law according to which we have a plus b into c the whole is equal to a plus e the whole into a plus c where these a b and c are the elements of the set B. So using this distributed law for this expression that is x plus x complement into y we get this is equal to x plus x complement the whole into x plus y the whole into this expression that is x complement plus x into y the whole. So this is using the distributed law for the element a which belongs to the set B there exists an inverse a complement such that we have a plus a complement is equal to 1 which is same as a complement plus a. So x plus x complement would be equal to 1 so this is equal to 1 into x plus y the whole this whole into x complement plus x into y the whole. This is since a plus a complement is equal to 1 so here we get 1. Now we have the identity element 1 for the operation of the product where we have a into 1 is equal to a which is same as 1 into a. So here we have 1 into x plus y the whole is x plus y this whole into x complement plus x into y the whole now for this expression also we would use the distributed law. So using the distributed law here we get x complement plus x the whole into x complement plus y the whole that is here we have used the distributed law. Now again this x plus x complement would be 1 so this is equal to x plus y the whole into 1 into x complement plus y the whole. Now further we would get this is equal to x plus y the whole into x complement plus y the whole that is 1 into x complement plus y would be x complement plus y. Now here we would use this distributed law so this would be equal to y plus x into x complement now we also have an inverse of the element a which is a complement such that a into a complement is equal to 0 which is same as a complement into a where the 0 is the identity element for the operation of sum. So this would be equal to y plus 0 so if we have a into a complement is equal to 0 and so we get this is equal to y therefore finally we have x plus x complement into y the whole into x complement plus x into y the whole is equal to y. This is what we were supposed to prove so in this session hope you understood the solution of this question.