 Hello and welcome to the session. In this session we discussed the following question that says simplify the Boolean expression A into A plus V the whole plus B complement plus A the whole into B the whole complement. Let's now proceed with the solution. The given Boolean expression is A into A plus V the whole plus B complement plus A the whole into B the whole complement. And we are supposed to simplify this Boolean expression which could be done by using the basic axioms and the important theorems. If we have this Boolean algebra then according to the distributive law we have A into B plus C the whole would be equal to A into B plus A into C where A, B and C are the elements of the set B. This is the distributive law. Now using this distributive law for this expression that is A into A plus V the whole we get this equal to A into A plus A into B plus this expression remains as it is that is B complement plus A the whole into B the whole complement. So we have used here the distributive law. Now for the idempotent law we have A into A is equal to A. So for A into A we would write A plus A into B plus for this expression we will use the distributive law and so here we can write B complement into B plus A into B the whole complement. This is using the idempotent law for this expression and the distributive law this expression. Now for the element A where exists its inverse A complement such that A into A complement would be same as A complement into A and this would be equal to 0 which is the identity element for the operation of sum. So using this we get B complement into B and 0 so this is equal to A plus A into B plus 0 plus A into B the whole complement this is since B complement into B is equal to 0 or this is further equal to A plus A into B plus A into B the whole complement. Now we have the demotent law which says that A into B whole complement is equal to A complement plus B complement so applying the demotent law for this expression we have this is equal to A plus A into B plus A complement plus B complement this is by D demotent law we can again apply the distributive law for this expression that is this distributive law so we get this is equal to A into 1 plus B the whole plus A complement plus B complement. So here we have used the distributive law now we have a theorem according to which we have A plus 1 is equal to 1 now this can be proved as 1 is equal to A plus A complement since we know that for any element A where exists inverse A complement such that A plus A complement is equal to 1 now we can further write this as A plus A complement into 1 this is using the identity that is A complement into 1 is equal to A complement now we will use the distributive law so using the distributive law we have A plus A complement the whole into A plus 1 now A plus A complement is 1 into A plus 1 and 1 into A plus 1 the whole is equal to A plus 1 so we have A plus 1 is equal to 1 so using this result we can write here A into 1 that is instead of 1 plus B we have 1 plus A complement plus B complement we have the identity element for the operation of the product as 1 such that A into 1 is equal to 1 into A is equal to A so using this we have A into 1 is equal to A plus A complement plus B complement this is since we have used A into 1 is equal to A now A plus A complement would be equal to 1 so here we have 1 plus B complement and since we had shown that A plus 1 is equal to 1 so we can say that 1 plus B complement is equal to 1 so we have simplified the given expression A into A plus B the whole plus E complement plus A the whole into B the whole complement equal to 1 so this is the answer let us complete this session hope you understood the solution of this question