 Hello and welcome to the session. In this session, we discuss the following question that says simplify the Boolean expression x into x plus y the whole plus y compliment plus x the whole into y the whole. Let's proceed with the solution now. The given Boolean expression is x into x plus y the whole plus y compliment plus x the whole into y the whole. For this Boolean algebra, the product operation is commutative. That is, we have a into b is equal to b into a. This is the commutative law. So, applying this commutative law for this expression, we have this is equal to x into x plus y the whole plus y into y compliment plus x the whole. This is by the commutative law. Now, by the distributive law, we have a into b plus c the whole is equal to a into b plus a into c. Now, applying this distributive law for these two terms of this expression, we get this is equal to x into x plus x into y plus y into y compliment plus y into x. Here, we have used the distributive law. Further, we can write x into x as x plus x into y would remain as it is plus now y into y compliment would be 0 as for any element a, there exists its inverse a compliment such that a into a compliment is same as a compliment into a which is equal to 0 like the identity element for the operation of sum. So, y into y compliment is 0 plus y into x remains as it is. So, here we have used that a into a is equal to a and a into a compliment is equal to 0. Now, plus x into y could be written as x into 1 plus y plus we have used this distributive law here plus y into x remains as it is. So, here we have used the distributive law. Now, a plus 1 could be proved as equal to 1, let us see how now 1 can be written as a plus a compliment that is for any element a we have its inverse a compliment such that a plus a compliment is equal to 1 which is the identity element for the operation of the product. Now, further we can write a plus a compliment into 1 since we know that a into 1 is equal to a where this 1 is the identity element for the operation of the product. So, we have written a compliment as a compliment into 1. Now, next we will use the distributive law to get a plus a compliment the whole into a plus 1 the whole. Now, a plus a compliment is equal to 1. So, 1 into a plus 1 the whole which is equal to a plus 1. So, we have a plus 1 is equal to 1. So, here we can write x into 1 that is 1 plus y could be written as 1 plus y into x. Now, using this commutative law for y into x we write it as x into y and this term x into 1 can be written as x as we know that 1 is the identity element for the operation of the product. So, a into 1 is same as 1 into a is equal to a. So, x into 1 is x. Now, using the distributive law here we can write this as x into 1 plus y the whole this is using the distributive law then we have a plus 1 is equal to 1. So, 1 plus y could be written as 1. So, x into 1 or x into 1 would be same as x. So, thus we get x. So, we have simplified the given whole in expression x into x plus y the whole plus y compliment plus x into y the whole is equal to x. So, this is the answer. This comes easy session hope you have understood the solution of this question.