 Hello and welcome to the session. Let us discuss the following problem today. Show that the relation r in the set a is equal to x belongs to z such that 0 is less than equal to x is less than equal to 12 given by r is equal to a comma b such that a is equal to b is an equivalent solution. Find the set of all elements related to 1. Now let us write the solution given to us as r is equal to a comma b such that a is equal to b. Now a is equal to 0 comma 1 comma 2 so on till 11 comma 12. Now let us check for reflexivity. Clearly for 1 belongs to a 1 comma 1 belongs to r as r contains a b where a is equal to b. Therefore r is reflexive. Now let us check for symmetry. For 1 belongs to a 1 comma 1 belongs to r which is clearly symmetric. Therefore r is symmetric. Now let us check for transitivity. We have 1 comma 1 belongs to r, 2 comma 2 belongs to r and so on. Let us consider 1 comma 1 belongs to r. Now the only ordered pair whose first element is 1 is 1 comma 1. Therefore for 1 comma 1 belongs to r and 1 comma 1 belongs to r we have 1 comma 1 belongs to r because for a b belongs to r and b c belongs to r we have a c belongs to r for transitivity and here a is equal to b is equal to c is equal to 1. Therefore r is transitive. Now since r is reflexive r is symmetric and r is transitive hence r is an equivalence relation. Now let us do the second part. Let x be an element of a such that x comma 1 belongs to r then by definition of r is equal to a comma b such that a is equal to b we have x is equal to 1. Hence the set of all elements of a which are related to 1 is single term 1. I hope you understood this problem. Bye and have a nice day.