 Hello and welcome to the session. In this session we discuss the following question which says, if the set B and the binary operations sum, product and complement is a Boolean algebra, prove that first x complement plus y complement, the whole complement is equal to x into y for all x and y belongs to the set B, second x complement into y complement, the whole complement is equal to x plus y for all x, y belongs to the set B. Before we move on to the solution, let's discuss the D Morgan's law. According to this we have A plus B, the whole complement is equal to A complement into B complement and A into B, the whole complement is equal to A complement plus B complement. This is the key idea that we use for this question. Let's proceed with the solution now. Now in the first part we need to prove that x complement plus y complement, the whole complement is equal to x into y where we have these x and y are the elements of the set B. So for this, first we consider x complement plus y complement, the whole complement. Now this would be equal to x complement, the whole complement into y complement, the whole complement. This is using the D Morgan's law that is we have used this law A plus B, the whole complement is equal to A complement into B complement. Now x complement, the whole complement is equal to x into y complement, the whole complement is equal to y. So where we have x into y, since we know that A complement, the whole complement is equal to A. So we now have x complement plus y complement, the whole complement is equal to x into y. So we have this first part. Now let's move on to the second part in which we are supposed to prove x complement into y complement, the whole complement is equal to x plus y where again this x and y are the elements of the set B. So to prove this, we first consider x complement into y complement, the 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. So using this D Morgan's law, we would get this is equal to x complement, the whole complement plus y complement, the whole complement. This is using the D Morgan's law. So x complement, the whole complement is x plus y complement, the whole complement is y. Since we know that A complement, the whole complement is equal to A. So we now have x complement into y complement, the whole complement is equal to x plus y. Hence we have proved the second part also. So this completes the session. Hope you have understood the solution of this question.