 Hello and welcome to the session. Let us discuss the following problem today. Consider the binary operations star from r cross r to r and o from r cross r to r defined as a star b is equal to mod of a minus b and a o b is equal to a for all a b belongs to r. Show that star is commutative but not associative but o is associative but not commutative. Show that for all a b c belongs to r a star b o c is equal to a star b o a star b. Thus o distribute over star. Justify your answer. Now let us write the solution. Given to us the binary operations as star from r cross r to r defined as a star b is equal to mod of a minus b for all a b belongs to r. And function o from r cross r to r defined as a o b is equal to a for all a b belongs to r. We have to show that star is commutative but not associative. Given to us a star from r b is equal to mod of a minus b for all a b belongs to r. Therefore, b star a is equal to mod of b minus a. Mod of a minus b is equal to mod of b minus a star b is equal to b star a and hence star is commutative. Next we have to prove for associativity. b belongs to r we prove that not equal to a star b star c and star 4 which is equal to 3 is not equal to 1. Therefore, 2 star 3 is not equal to 2 star 3 star 4 associative. We have proved that star is commutative but not associative. Now let us check for function o to show that creative commutative. Given to us is o is a function from r cross r to r such that a o b is equal to a. We observe that b o a is equal to b for all a b belongs to r. Not equal to b therefore a o b is not equal to b o a commutative. Let us show associativity. c belongs to r we have is equal to a o c which is equal to a is equal to a o b which is equal to a. Now since hence we have proved that o is associative but not commutative. Now let us check the distributivity. Now to show that r is distributive go proving it let us understand what is distributivity. A star is said to be distributive over plus in the set of real numbers r if a star b plus c is equal to a star b plus a star c for all a b c belongs to r. Now we solve it further o c is equal to a star b o a star c. We have to prove this is equal to a star b which is equal to mod of a minus b mod of a minus b o mod of a minus c which is equal to mod of a minus b. A star b is equal to mod of a minus b and a star c is equal to mod of a minus c thus a star b o c is equal to a star b o a star c. Therefore distributive distributive over o and hence o c is equal to a star b o a star c. Now we will check whether o distributes over star. For any a b c belongs to r we have r c which is equal to a o mod of b minus c which is equal to a which is equal to mod of a minus a which is equal to 0. Thus a o b star c is not equal to a o b star a o c not distributive over. Therefore our answer is no. I hope you understood the problem bye and have a nice day.