Recover a finite set from A × A size and sample pairs: Given that A × A has 9 elements and includes (−1, 0) and (0, 1), determine A.

Introduction / Context:If |A × A| = 9, then |A|^2 = 9, giving |A| = 3. The given pairs show elements actually present in A. Combining cardinality with observed components identifies A uniquely up to ordering of elements (sets are unordered).

Given Data / Assumptions:

• |A × A| = 9 ⇒ |A| = 3
• (−1, 0) ∈ A × A ⇒ −1 ∈ A and 0 ∈ A
• (0, 1) ∈ A × A ⇒ 0 ∈ A and 1 ∈ A

Concept / Approach:Collect the distinct elements witnessed in coordinates: {−1, 0, 1}. This already has size 3, matching the required cardinality. Therefore A must be {−1, 0, 1}.

Step-by-Step Solution:From (−1,0) and (0,1), elements −1, 0, 1 all occur|A| must be 3Thus A = {−1, 0, 1}

Verification / Alternative check:Compute |A × A| with A = {−1,0,1}: 3^2 = 9, consistent with the problem statement.

Why Other Options Are Wrong:They have fewer than 3 elements and cannot produce 9 ordered pairs in A × A.

Common Pitfalls:Forgetting that the presence of a coordinate in any pair certifies membership in A; both coordinates must come from A.

Final Answer:A = {−1, 0, 1}