The Cartesian product A × A has 9 elements among which are found…

Question 10 of 33advanced🔍 AnalyzeLong Answer4 marks

The Cartesian product $A \times A$ has 9 elements among which are found $(-1, 0)$ and $(0, 1)$ . Find the set $A$ and the remaining elements of $A \times A$ .

Correct Answer

A = \{-1, 0, 1\}

Remaining elements of $A \times A$ are:

(-1, -1),\; (-1, 1),\; (0, -1),\; (0, 0),\; (1, -1),\; (1, 0),\; (1, 1)

Exercise: EXERCISE 2.1 | Q: 10 | (Page 4)

For More Understanding

Explanation

This question tests understanding of Cartesian products and the relationship between the number of elements in a set and its Cartesian product. The key insight is that $n(A \times A) = n(A) \times n(A)$ , so if $A \times A$ has 9 elements, then $n(A) = 3$ . From the given ordered pairs, we extract all elements appearing in either position to determine $A$ completely.

Solution Steps

Step 1: Since $n(A \times A) = n(A) \times n(A) = 9$ , we get $n(A)^2 = 9$ , therefore $n(A) = 3$ .
Step 2: From the ordered pairs $(-1, 0)$ and $(0, 1)$ , extract elements: $-1, 0, 1$ . Since $A$ has exactly 3 elements, $A = \{-1, 0, 1\}$ .
Step 3: Form $A \times A = \{(-1,-1),\; (-1,0),\; (-1,1),\; (0,-1),\; (0,0),\; (0,1),\; (1,-1),\; (1,0),\; (1,1)\}$ .
Step 4: Remove given pairs $(-1, 0)$ and $(0, 1)$ to get remaining 7 elements.

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