Let A be a set containing 10 elements. Suppose A is partitioned into 3 non-empty disjoint subsets A₁, A₂, and A₃ . A relation R is defined on A such that (x, y) R if and only if both x and y belong to the same subset A_i for some 1 i 3 . The minimum possible number of ordered pairs in the relation R is

Step-by-step solution

The relation R is defined such that (x, y) R x, y A_i . This means R = (A₁ A₁) (A₂ A₂) (A₃ A₃) . Since A₁, A₂, A₃ are disjoint, the number of ordered pairs in R is given by n(R) = |A₁|^2 + |A₂|^2 + |A₃|^2 . Let x, y, z be the number of elements in A₁, A₂, A₃ respectively. We are given x + y + z = 10 , where x, y, z 1 are integers. To minimize x^2 + y^2 + z^2 , the values of x, y, z should be as close to each other as possible. Thus, we choose x = 3, y = 3, z = 4 . The minimum number of ordered pairs is 3^2 + 3^2 +