Question

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.​

Answer 1

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➡️We know that,

➡️If n(A) = p and n(B) = q, then n(A × B) = pq.

➡️Also, n(A × A) = n(A) × n(A)

➡️Given,

➡️n(A × A) = 9

➡️So, n(A) × n(A) = 9

➡️Thus, n(A) = 3

➡️Also given that, the ordered pairs (–1, 0) and

➡️(0, 1) are two of the nine elements of A × A.

➡️And, we know in A × A = {(a, a): a ∈ A}.

➡️Thus, –1, 0, and 1 has to be the elements of A.

➡️As n(A) = 3, clearly A = {–1, 0, 1}.

➡️Hence, the remaining elements of set A × A are as follows:

➡️(–1, –1), (–1, 1), (0, –1), (0, 0), (1, –1), (1, 0), and (1, 1)

Answer 2

From the product, it has elements 1, 0, -1. Since there are nine elements, set A consists of 3 elements. Now, set A has elements 1, 0, -1.

It is clear the seven remaining elements are (1,1), (1,0), (1,-1), (0,0), (0,-1), (-1,1), (-1,-1).