The Cartesian product has 9 elements among which are found and . Find the set and the remaining elements of .
A = {-1, 0, 1}
Remaining elements of A × A are: (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), (1, 1)
Explanation
This question tests understanding of Cartesian products and the relationship between the number of elements in a set and its Cartesian product. From the context, we know that if n(A) = p and n(B) = q, then n(A × B) = pq. Since A × A has 9 elements, we get n(A)² = 9, so n(A) = 3. The given ordered pairs (-1, 0) and (0, 1) tell us that -1, 0, and 1 must all be elements of A. Since A has exactly 3 elements, A = {-1, 0, 1}. The complete Cartesian product A × A contains all 9 ordered pairs formed by combining these three elements.
Solution Steps
Step 1: Given n(A × A) = 9. Since n(A × A) = n(A) × n(A), we have n(A)² = 9, so n(A) = 3.
Step 2: From the ordered pair (-1, 0) ∈ A × A, both -1 and 0 must be elements of A.
Step 3: From the ordered pair (0, 1) ∈ A × A, both 0 and 1 must be elements of A.
Step 4: Therefore, A = {-1, 0, 1} (exactly 3 elements).
Step 5: A × A = {(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 0), (0, 1), (1, -1), (1, 0), (1, 1)}.
Step 6: Remaining elements (excluding (-1, 0) and (0, 1)) are: (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), (1, 1).