 Hi and welcome to the session. I am Shashi. Let us do one question. Question is, for each binary operation defined below, determine whether star is commutative or associative. Sixth part is on, on minus minus one, where r is the set of all real numbers, define a star b is equal to a upon b plus 1. First of all, let us understand the key idea to solve the given question. A binary operation star on set a is commutative if a star b is equal to b star a for every a b belonging to a. 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 bracket b star c for every a b c belonging to set a. Let us now start with the solution. We know on set r minus minus one, where r is the set of all real numbers. Binary operation star is defined as a star b is equal to a upon b plus one. First of all, let us check if star is commutative. We know a star b is equal to a upon b plus one. Now, let us find b star a, b star a we know is equal to b upon a plus one. Now, we know a plus one is not equal to b upon a plus one. Every a b belonging to r minus minus one, this implies a star b is not equal to b star a. Now, we can write binary operation star is not commutative on set r minus minus one, where r is the set of all real numbers. Now, let us find if star is associative. First of all, we will find a star b star c we know it is equal to a upon b plus one star c. For every a b c belonging to minus minus one, this is counter equal to a upon b plus one upon c plus one. We know this is bracket is our a and this is our b. We know a star b is equal to a upon b plus one. So, this is our a and this is our b. So, a upon b plus one. Now, this is called the equal to a upon b plus one multiplied by c plus one. So, we can write a star b star c is equal to a upon b plus one upon c plus one. Let us now find out a star bracket b star c. It is further equal to a star b upon c plus one right. Now, this is our a and this bracket is our b. So, we will write a upon this is b b plus one. So, b upon c plus one plus one. Now, simplifying we get multiplied by c plus one upon b plus c plus one. So, we get a star bracket b star c is equal to a multiplied by c plus one upon b plus c plus one. So, we know a upon b plus one multiplied by c plus one is not equal to a multiplied by c plus one upon b plus c plus one. For every a b c belonging to r minus minus one where r is the set of all clear numbers. Now, we know a upon b plus one multiplied by c plus one is equal to a star b star c. So, we substitute a star b star c in place of a upon b plus one multiplied by c plus one right and we know a multiplied by c plus one upon b plus c plus one is equal to a star b star c. So, substituting its value we get a star bracket b star c. So, we get a star b star c is not equal to a star bracket b star c. Therefore, we can write binary operations star is not associative on r minus minus one where r is the set of all real numbers. So, our final answer is star is neither commutative nor associative. This completes the session. Hope you understood the session. Take care and goodbye.