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प्रश्न
The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0, 1). Find the set A and the remaining elements of A × A.
उत्तर
We know that if n (A) = p and n(B) = q, then n (A × B) = pq.
∴ n (A × A) = n (A) × n (A)
It is given that n (A × A) = 9
∴ n (A) × n (A) = 9
⇒ n (A) = 3
The ordered pairs (–1, 0) and (0, 1) are two of the nine elements of A × A.
We know that A × 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 × A are (–1, –1), (–1, 1), (0, –1), (0, 0), (1, –1), (1, 0), and (1, 1)
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