 Hi, and welcome to the session. I am Shashi. Let us do one question. Question is, let A is equal to n cross n and star with a binary operation on A defined by AB star CD is equal to A plus C B plus D. Here n is the set of all natural numbers. Show that star is commutative and associative. Find the identity element for star on A if any. First of all, let us understand the key idea to solve the given question. A binary operation star on set X is called commutative if A star B is equal to B star A for every AB belonging to set X. Let us now define associative. A binary operation star from A cross A to A is said to be associative if A star bracket B star C is equal to A star B star C for every ABC belonging to set A. Now, next is the definition of identity for operation. Given a binary operation star from A cross A to A and element E belonging to A if it exists is called identity for operation star if A star E is equal to A is equal to E star A for every A belonging to set A. Let us now start with the solution. We are given in the question let A is equal to n cross n where n is the set of natural numbers and star B the binary operation on A defined by AB star CD is equal to A plus C B plus D. First of all, let us take if star is commutative. We are given AB star CD is equal to A plus C B plus D. Now we will find CD star AB. It is equal to C plus A and D plus B can be written as A plus C B plus D as addition is commutative. So, C plus A can be written as A plus C and D plus B can be written as B plus D. Now, we can see A plus C B plus D is equal to AB star CD. So, we can write AB star CD is equal to CD star AB. So, from key idea we find star is commutative. Now let us show that star is associative for every AB CD EF belonging to set of natural numbers. Let us consider AB star bracket CD star EF. Now this is further equal to AB star bracket C plus E comma D plus F. Now this is further equal to A plus C plus E B plus D plus F. Right? Now let us consider AB star CD star ES. Now this is further equal to A plus C B plus D star EF which is equal to A plus C plus E B plus D plus F. Right? Now let us name this equation as 1 and this equation as 2. Now from 1 and 2 we can see AB star CD star EF is equal to AB star CD star EF from 1 and 2 we get the result AB star CD star EF is equal to bracket AB star CD bracket close star EF. From key idea we already know a binary operation is associative if A star bracket B star C is equal to A star B star C. So using the key idea we can see the given operation is associative. So we can write binary operation star is associative. Now the next part of the question is to find that if star has identity element let ordered pair XY be the identity element in set A. Then ordered pair AB star ordered pair XY is equal to ordered pair AB for every ordered pair AB belonging to set A. We know from key idea that A star E must be equal to A must be equal to E star A. So A star E this is our identity element so A star E must be equal to A that's what we have used here. Now this implies A plus X B plus Y is equal to ordered pair AB for every ordered pair AB belonging to set A where set A is equal to N cross N. N is the set of all natural numbers. Now we can write this implies A plus X must be equal to A and B plus Y must be equal to B. This further implies X is equal to 0 and Y is equal to 0. But this is not possible since XY belong to set of natural numbers and set of natural numbers does not contain 0. Therefore identity element does not exist for binary operation star. So this is our required answer. This completes the session. Hope you understood the session. Take care and goodbye.