 Hi, welcome to the session. I am Asha and I am going to help you with the following question which says in each of the following, determine whether the statement is true or false. If it is true, prove it and if it is false, give an example. So the first one is, if X belongs to A and A belongs to B, then X belongs to B. So the given statement is false since X and Y and B is a set having elements Z and XY. Then from here we find that A belongs to B but do not belong to B. But here it is given that if X belongs to A and A belongs to B, then X belongs to B. But here we have X belongs to A and A belongs to B but still X do not belong to B. So this contradicts the given statement and hence the statement is false. So this is the first part and now proceeding on to the second part where we have if A is contained in B and B belongs to C, then A belongs to C. So this statement is false and let us take one example to show that let A be any set having element A and B be a set having elements A, B and C and C be a set having elements D and the set ABC. And from here we find that A is a subset of the set having elements ABC belongs to C but the element A do not belong to the set C rather A do not belong to C. So this contradicts the given statement that if A is a subset of B and B belongs to C then A belongs to C. So the given statement is false. Let us now proceed on to the third part which says if A is a subset of B and B is a subset of C then A is a subset of C. This statement is true and let us prove it. So X be any element which belongs to A. Now since A is a subset of B so all the elements which belong to A will belong to B. So this implies that X belongs to B and again since B is a subset of C so all those elements which belong to B must belong to set C also. So this implies X belongs to C fine. So X is any arbitrary element of set A so this implies all the elements which are in set A must belong to set C which implies A is a subset of C and thus the given statement is true. So this completes the third part and now proceeding on to the fourth part which says if A is not a subset of B and B is not a subset of C then A is also not subset of C. So this statement is false and let us take one example to prove it. Let A be a set having elements 1 and 2, B be a set having elements 3 and 4 and C be a set having elements 1, 2 and 3. Now from here we can see that 1 and 2 do not belong to B hence A is not a subset of B and 4 do not belong to set C. Therefore B is also not a subset of C but the elements of A which are 1 and 2 belong to the set C this implies A is a subset of C and we are given that if A is not a subset of B and B is not a subset of C then A is not a subset of C which contradicts the example and hence the given statement is false. Now proceeding on to the fifth part which says if X belongs to A, A is not a subset of B then X belongs to B. So this statement is false let us take one example to prove it. Let A be any set having elements X and Y and B be a set having elements Y and Z. Now X belong to A, observing we find that elements of A are X and Y and elements of Z be a Y and Z. So this implies A is not a subset of B thus X belong to B is not true. The given statement is false. So this completes the fifth part and now proceeding on to the last part which says A is a subset of B, X is an element which do not belong to B then X do not belong to A. This statement is true is a subset of B implies all elements of A belong to B also clearly Y belonging to B we must have Y belonging to A the element which do not belong to B can never belong to a subset A. So if X do not belong to B then this implies X do not belong to A also and hence the given statement is true. So this completes the last part and hence the session hope you enjoyed it take care and have a good day.