 Hi and welcome to the session I am Shashi and I am going to help you to solve the following question. Question is let star be a binary operation on set Q of rational numbers as follows. Sixth part is a star b is equal to a b square, find which of the binary operations are commutative and which are associative. First of all let us understand the key idea to solve the given question. A binary operation star on the set X is called commutative if a star b is equal to b star a for every a b belonging to set X. Now 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 Q where Q is the set of all rational numbers. Binary operation star is defined as a star b is equal to a b square first of all let us take whether binary operation star is commutative now we know a star b is equal to a b square then b star a would be equal to b a square but a b square is not equal to b a square so this implies a star b is not equal to b star a so we can write binary operation star is not commutative on Q where Q is the set of all rational numbers. Now let us check if star is associative for a b c belonging to set of rational numbers we have a star b star c is equal to a b square star c. Now this is further equal to a b square c square we know a star b is equal to a b square so this is our a and we get c square right so our a star b star c is equal to a b square c square now let us find out a star bracket b star c it is equal to a star b c square right it is further equal to a b c square whole square we know this is our a and this bracket is our b and we know a star b is equal to a b square so we get a b square c raised to the power 4 but a b square c square is not equal to a b square c raised to the power 4 so this implies a star b star c is not equal to a star bracket b star c so we can write binary operation star is not associative on set Q where Q is the set of all rational numbers so our final answer is binary operation star is neither commutative nor associative on set Q where Q is the set of all rational numbers so this completes the session hope you understood the session take care and goodbye