Question

The Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A x A.

Answer 1

We know that if n(A) = p and n(B) = q, then n(A x B) = pq.
∴ n(A x A) = n(A) x n(A)
It is given that n(A x A) = 9
∴ n(A) x n(A) = 9
⇒ n(A) = 3
The ordered pairs (-1, 0) and (0, 1) are two of the nine elements of A x A.
We know that A x A = {(a, a): a ∈A}. Therefore, -1, 0, and 1 are elements of A.
Since n(A) = 3, it is clear that A = {-1, 0, 1}.
The remaining elements of set A x A are (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), and (1, 1)