Set Theory & Algebra

A binary operation [Tex]\oplus[/Tex] on a set of integers is defined as x [Tex]\oplus[/Tex] y = x² + y². Which one of the following statements is TRUE about [Tex]\oplus[/Tex]?

Consider the set S = {1, ω, ω2}, where ω and w² are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

For the composition table of a cyclic group shown below

Which one of the following choices is correct?

If P, Q, R are subsets of the universal set U, then 

Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:

Consider the set S = (A, B, C, D}. Consider the following 4 partitions π1, π2, π3, π4 on S : π1 = {ABCD}'  , π2 = {AB}' , {CD}' , π3 = {ABC}' , {D}' ,  π4 = A' , B' , C' , D'.  Let P be the partial order on the set of partitions S' = {π1, π2, π3, π4} defined as follows : πi P πj if and only if πi refines πj. The poset diagram for (S', P ) is :