Question

if AXA has 9 elements among which are found (-1,0) and (0,1) find set a ?
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Answer 1

Let n(A) = p.

Given that n(A * A) = 9

= > n(A) * n(A) = 9

= > p * p = 9

= > p^2 = 9

= > p = 3.

Clearly, Set A has 3 elements.

Since(-1,0) ∈ A * Aand (0,1) ∈ A * A.

= > (-1,0) ∈ A and 0 ∈ A

= > (0,1) ∈ A and 1 ∈ A

Therefore -1, 0, 1 ∈ A.

Hence A = {-1, 0, 1}.

Therefore remaining elements are :

{-1,-1}, { -1,1}, {0,-1}, {0,0}, {1,-1}, {1,0}, {1,1}.

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Answer 2

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We know that,

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

From the given,

n(A × A) = 9

n(A) × n(A) = 9,

n(A) = 3 ……(i)

The ordered pairs (-1, 0) and (0, 1) are two of the nine elements of A × A.

Therefore, A × A = {(a, a) : a ∈ A}

Hence, -1, 0, 1 are the elemets of A. …..(ii)

From (i) and (ii),

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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