 Following welcome to the session. In this session we discussed the following question that says in the Boolean algebra consisting of the set B and the operations sum product and complement show that x into y plus x plus y complement the whole into y, the whole complement is equal to 1 where these x and y are the elements of the set B. Let's proceed with the solution now. We are supposed to prove x into y plus x plus y complement the whole into y, the whole complement is equal to 1 and these x and y are the elements of the set B. So, we consider x into y plus y complement the whole into y, the whole complement. Now if we have this Boolean algebra then by the D Morgan's law we have a into b the whole complement is equal to a complement plus b complement. This is the D Morgan's law. So, applying the D Morgan's law in this expression that is a into b the whole complement is equal to a complement plus b complement we get this is equal to x into y plus y complement the whole complement plus y complement the whole. This is using the D Morgan's law. Now we have another D Morgan's law which is a plus b the whole complement is equal to a complement into b complement. Now applying this C Morgan's law for this expression we would get this equal to x into y plus x complement into y complement the whole complement plus the term y complement remains as it is. This is again by the D Morgan's law. Now this would be further equal to x into y plus x complement into y complement the whole complement would be y plus y complement the whole. So, here we have used that a complement the whole complement is equal to a. Now we know that the sum operation is associative so we have a plus b plus c the whole would be equal to a plus b the whole plus c. This is the associative law. Now using the associative law here we get this is equal to x into y plus x complement into y the whole plus y complement. This is using the associative law. Now further the product operation is commutative so we have a into b is equal to b into a this is the commutative law. So applying this commutative law for these two terms separately we have this is equal to y into x plus y into x complement the whole plus y complement. This is by the commutative law. Now we have a distributive law where we have a into b plus c the whole is equal to a into b plus a into c. This is the distributive law. Now using this distributive law for this expression we have this is equal to y into x plus x complement the whole plus y complement remains as it is. So this is using the distributive law. For an element a which belongs to the set b we have its inverse a complement such that a plus a complement is equal to 1 and this is same as a complement plus a and this 1 is the identity element for the operation of the product. So using this we have x plus x complement is 1 so this is equal to y into 1 plus y complement. This is because a plus a complement is equal to 1. Also we have an identity element 1 for the operation of the product such that a into 1 is equal to a which is same as 1 into a. So this y into 1 would be equal to y plus y complement remains as it is. This is because we have a into 1 is equal to a. Now we already know that a plus a complement is 1 so y plus y complement would be equal to 1 thus we now have into y plus y complement the whole into y the whole complement is equal to 1. This is what we were supposed to prove so hence proved the question hope you understood the solution of this question.