Let A = 0,1,2, , 9 . Let R be a relation on A defined by (x, y) R if and only if |x-y| is a multiple of 3. Given below are two statements: Statement I : n( R )=36 . Statement II : R is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below

Step-by-step solution

The relation R on A = 0,1,2, ,9 is defined by (x,y) R iff |x-y| is a multiple of 3, i.e., x y 3 . Statement II: R is reflexive ( |x-x|=0 , a multiple of 3), symmetric ( |x-y|=|y-x| ), and transitive (if x y and y z 3 , then x z 3 ). So R is an equivalence relation. Statement II is correct. Statement I: The equivalence classes are 0,3,6,9 (4 elements), 1,4,7 (3 elements), 2,5,8 (3 elements). n(R) = 4^2 + 3^2 + 3^2 = 16 + 9 + 9 = 34 36 . Statement I is incorrect.