 Hello and welcome to the session. In this session we discuss the following question which says, let the set B containing the elements A and B and let operation of sum and product be defined as follows. As shown in these two tables, show that the system consisting of the set B and the operations of sum and product is not a Boolean algebra. For a system consisting of the set B operations of sum and product to be a Boolean algebra, the following laws must be satisfied. First is the closure law, the commutative law, the associative law, the distributive law, then we have the inverse law, identity law. If these six laws are satisfied, then this system would be a Boolean algebra. This is the key idea that we use for this question. Let's proceed with the solution now. We have a set B which contains the elements A and B and these tables shows how the operations of sum and product are defined. Now first of all, we consider the closure law and see if this closure law is satisfied or not. Now we have A and B are the elements of the set B. Now consider A plus B, from this table, we find that A plus B is A and this also belongs to the set B. Then A into B, from the table above, we have A into B is B and this B also belongs to the set B. Therefore, as A and B belongs to the set B, A plus B also belongs to the set B and A into B also belongs to the set B. Therefore, the closure law is satisfied. Next we move on to the commutative law. Again consider A and B to be two elements of the set B. Now A plus B is equal to A. Now B plus A is equal to A. So these two are equal. That is we have A plus B is equal to B plus A. Now consider A into B. This is equal to B and B into A is also equal to B. So they are also equal. That is A into B is equal to B into A and thus we can say that the commutative law is satisfied. Next we have the associative law. For this we take the elements A and B belonging to the set B. Now consider A plus A plus B the whole. Now from this table we have A plus B is A. This is equal to A plus A and from this table again we have A plus A is B. So this is equal to B. Now we consider A plus A the whole plus B. Now we had observed A plus A is B plus B. Now from this table we have B plus B is A. So this is equal to A. Now from these two we observe that A plus A plus B the whole is not equal to A plus A the whole plus B. Now consider A into A into B the whole. This is equal to. Now from this table we have A into B is B. So this is equal to A into B. That is the place of A into B we write B. Now again A into B is B. Let's consider A into A the whole into B. So from this table we have A into A is B. So now B into B is equal to A. So these two are not equal. That is A into A into B the whole is not equal to A into A the whole into B. And therefore we can say that the associative law is not satisfied. Hence we can conclude that the system consisting of the set B and the operations of sum and product not a Boolean algebra is the session. Hope you have understood the solution of this question.