 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says show that the following four conditions are equivalent. First is A is a subset of B, second is A minus B is equal to an empty set, third is A union B is equal to B and fourth is A in the section B is equal to A. So first let us learn what do we mean by a subset. This denotes that A is a subset of B and A is a subset of B if every element of A is also an element of set B. Having all those X, X belong to A and X do not belong to B. So with these two definitions we will show that the four conditions are equivalent. This is a key idea. Let us now start with the solution and we will start with the first part and show that it is equivalent to the second part. Second is equivalent to third and third is equivalent to four and fourth in turn is again equivalent to part one. So starting with the first one we will show that one implies two. Now one says A is a subset of B. So let us interpret this in the form of a when diagram. Suppose this is set B and this is set A. So this denotes A and this whole denotes set B. Now from the when diagram we observe that set B is a subset of B that is all elements of A belong to B also and it is contained in B. Now second condition says A minus B has no elements that is A minus B is equal to five. This means all elements of A also elements of B and it can also be seen from the when diagram. All the elements which are in A are also elements of set B and this implies that A is a subset of B. Now the third condition is union of and B contains all elements of A or B or both. Now from the when diagram we see that A union B contains the same elements as B. So this implies that all the elements of A belong to B and hence A is a subset of B. Now the fourth condition says intersection of A and B contains common elements A and B. This implies A intersection B is equal to A implies that common elements are elements of A subset of B. Now four conditions confirm the same idea that the elements of set A are also the elements of set B hence the given conditions are equivalent. So this completes the solution. Hope you enjoyed it. Take care and have a good day.