 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says show that for sets A and B, A is equal to A intersection B union A minus B and A union B minus A is equal to A union B. So let us begin with the solution and first let us show that X minus Y is equal to X intersection Y complement. So first let us take X belonging to the set X minus Y. So this implies X belong to X but X do not belong to Y. This implies X belongs to X and X belongs to Y complement which further implies that X belongs to X intersection Y complement which shows that X minus Y is a subset of X intersection Y dash. Now let us take X belonging to X intersection Y complement. This implies X belongs to X and X belongs to Y complement which further shows that X belongs to X and X do not belong to Y. Since it belongs to Y complement which implies that X belongs to X minus Y which shows that X intersection Y complement is a subset of X minus Y. Now from these two we can say that X minus Y is nothing but X intersection Y complement. So this we are going to use while solving the problem. Let us now begin to show that A is equal to A intersection B union A minus B. Now let us consider the right hand side which is A intersection B union A minus B and we will show that it is equal to the left hand side which is the set A. This can be written as A intersection B union A intersection B complement since X minus Y is equal to X intersection Y complement we have just shown and here in place of X we have A and in place of Y we have B. This can further be written as A intersection B union B complement and this is where the distributive law is equal to A intersection B union B complement is nothing but the universal set U and the intersection of any set with the universal set is the set itself which is A and this is the left hand side and this shows that A is equal to A intersection B union A minus B. Now let us show that A union B minus A is equal to A union B. So let us start with the left hand side which is A union B minus A. It can be written as A union B intersection A complement since X minus Y is equal to X intersection Y complement which we have shown and this can further be written as A union B intersection A union A complement. Which is further equal to A union B intersection the union of a set with its complement is the universal set and the intersection of any set with the universal set is the set itself. So we have A union B which is the right hand side and this shows that A union B minus A is equal to A union B. So this completes the solution hope you enjoyed it take care and have a good day.