 Hello and welcome to the session. In this session we discussed the following question which says find the complement of x plus x complement into y complement the whole into x complement plus x into y the whole. Let's proceed with the solution now. You have to find x plus complement into y complement the whole into x complement plus x into y the whole complement. First of all, we consider the expression x plus x complement into y complement the whole into x complement plus x into y the whole. For this Boolean algebra we have a distributive law according to which we have a plus b into c v whole is equal to a plus b v whole into a plus c v whole. This is the distributive law. So applying this distributive law for these two expressions we get this is equal to x plus x complement the whole into x plus y complement the whole this whole into now applying the distributive law to this expression we have x complement plus x the whole into x complement plus y the whole. This is using the distributive law. For any element a we have its inverse a complement such that a plus a complement is equal to 1 which is seen as a complement plus a where this 1 is the identity element for the operation of the product. So instead of x plus x complement we can write 1 so 1 into x plus y complement the whole into 1 into x complement plus y the whole. This is since a plus a complement is same as a complement plus a is equal to 1. We know that 1 is the identity element for the operation of the product. So we have a into 1 is equal to 1 into a is equal to a so 1 into x plus y complement the whole could be written as this y complement this whole into now 1 into x complement plus y the whole could be written as x complement plus y the whole. So therefore we now have x plus x complement into y complement the whole into x complement plus into y the whole is equal to x plus y complement the whole into x complement plus y the whole. That is we have simplified the given expression to this. Now we will find the complement of this given expression that is we need to find the complement of x plus x complement into y complement the whole into x complement plus x into y the whole complement and this would be equal to x plus y complement the whole into x complement plus y the whole and this whole complement. Now from 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 using this D Morgan's law to this expression we have this is equal to x plus y complement the whole complement plus x complement plus y the whole complement. We have another D Morgan's law which is a plus b the whole complement is equal to a complement into b complement. So from the D Morgan's law here we further have x complement into y complement the whole complement the whole plus x complement the whole complement into y complement the whole. So further this is equal to x complement into now y complement the whole complement is y plus x complement the whole complement is x into y complement this is since we have a complement the whole complement is equal to a. So this is equal to x complement into y plus x into y complement. Therefore we have x plus x complement into y complement the whole into x complement plus x into y the whole complement is equal to x complement into y plus x into y complement. So this is our answer this completes the session hope you have understood the solution of this question.