 Hi and welcome to the session. Today we will learn about binary operations. A binary operation star on a z A is a function from A cross A to A. We denote star A comma B that is binary operation on A comma B by A star B. Now a binary operation star on the set X is called commutative A star B is equal to B star A for every A comma B belonging to X. For example here star is a binary operation on set N given by A star B is equal to LCM of A and B and we need to show that the binary operation star is commutative. To show that binary operation star is commutative we will show that A star B is equal to B star A for any two natural numbers A and B. So A star B is equal to LCM of A and B and B star A is equal to LCM of B and A by the definition of A star B. Now LCM of B and A is same as LCM of A and B that means these both are equal. So this implies A star B is equal to B star A for every A B belonging to N that is the set of natural numbers. So from here we get that the binary operation star is commutative. Next a binary operation star from A cross A to A is said to be associative if A star B star C is equal to A star B star C for all A B C belonging to set A. Let's take the same example to show that the given binary operation is associative. So now we want to show that the binary operation star is associative. For this let us take three elements that is A B and C belonging to the set of natural numbers. So now A star B star C is equal to LCM of A B and C star B star C is equal to LCM of A B and C. So from this we get A star B star C is equal to A star B star C and this implies that the binary operation star is associative. Now given a binary operation cross A B belonging to set A if it exists is called identity for the operation star A star A is equal to A is equal to E star A for all A belonging to set A. Let's take one example here in this the identity of the binary operation star in the set of natural numbers. So here let us suppose that E is the identity element of the binary operation star in. So from this we get that A star E is equal to A is equal to E star A for all A belonging to N. Now let us consider A star E is equal to A for all A belonging to N and A star E will be equal to LCM of A and E by the definition of A star B. So we get LCM of A and E is equal to A for all A belonging to N and this is possible only when E is equal to 1. So from here we get E is equal to 1 and thus 1 is the identity element of the given binary operation. Lastly given a binary operation star from A cross A to A with the identity element in the set A an element A belonging to the set A is said to be invertible with respect to the operation star. There exists an element B in set A such that A star B is equal to E is equal to V star A. B is called the inverse of A and is denoted by A inverse. So now in the same example let us find out the elements of N which are invertible for the operation star. Let us suppose that A is the invertible element in the set N so that means there exists an element B belonging to the set N such that A star B is equal to E is equal to V star A. Here let us consider A star B is equal to E now A star B is the LCM of A and B. So from here we have LCM of A and B is equal to E and in this example we have obtained E equal to 1 so this will be equal to 1. And this is possible only when both A and B are equal to 1 itself. So this gives us A is equal to B is equal to 1. Now A is the invertible element in N and A is equal to 1 so that means 1 is the invertible element for the operation star in N. With this we finished this session hope you must have enjoyed it. Goodbye take care and keep smiling.