Understanding the Cartesian Product

The question asks about a Cartesian product A × A. The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where 'a' is an element of A, and 'b' is an element of B. In this case, the product is A × A, meaning it's the set of all ordered pairs (a, b) where both 'a' and 'b' are elements of the set A.

The number of elements in the Cartesian product A × A is given as 16. The cardinality (number of elements) of a Cartesian product is the product of the cardinalities of the individual sets. So, |A × A| = |A| × |A| = |A|².

Given that |A × A| = 16, we have |A|² = 16.

To find the cardinality of set A, we take the square root of 16. Since the cardinality must be a non-negative integer, |A| = \(\sqrt{16} = 4\). So, set A has exactly 4 elements.

Analyzing Statement 1: Possibility to Determine Set A

Statement 1 says it is possible to determine set A. We know |A| = 4. We are also given that the elements (0, 2) and (1, 3) are present in A × A.

According to the definition of the Cartesian product A × A, if an ordered pair (x, y) is in A × A, then 'x' must be an element of A, and 'y' must be an element of A.

• Since (0, 2) \(\in\) A × A, it means that 0 \(\in\) A and 2 \(\in\) A.
• Since (1, 3) \(\in\) A × A, it means that 1 \(\in\) A and 3 \(\in\) A.

From these two given elements, we know that the numbers 0, 1, 2, and 3 must all be elements of set A. So, the set {0, 1, 2, 3} is a subset of A ({0, 1, 2, 3} \(\subseteq\) A).

We previously determined that the cardinality of A is 4 (|A| = 4). We have found 4 distinct elements (0, 1, 2, and 3) that must be in A. Since A contains exactly 4 elements and we have identified 4 elements that it must contain, these must be the only elements in A.

Therefore, set A must be {0, 1, 2, 3}. We have successfully determined set A.

Statement 1 is correct.

Analyzing Statement 2: A × A Contains Element (3, 2)

Statement 2 asks if A × A contains the element (3, 2). Based on our analysis of Statement 1, we determined that the set A is {0, 1, 2, 3}.

The Cartesian product A × A is formed by taking all possible ordered pairs where the first element comes from A and the second element comes from A.

A × A = {(a, b) | a \(\in\) A and b \(\in\) A}

Using A = {0, 1, 2, 3}, the elements of A × A are:

To check if (3, 2) is in A × A, we need to see if the first element, 3, is in A and if the second element, 2, is in A.

From A = {0, 1, 2, 3}, we see that:

• 3 \(\in\) A (True)
• 2 \(\in\) A (True)

Since both elements 3 and 2 are in set A, the ordered pair (3, 2) is indeed an element of A × A.

Statement 2 is correct.

Conclusion on Statements

Both Statement 1 and Statement 2 are correct.

• Statement 1 is correct because knowing the cardinality of A × A and two specific elements allowed us to deduce the unique set A.
• Statement 2 is correct because once we determined A = {0, 1, 2, 3}, we could confirm that (3, 2) is an element of A × A as both 3 and 2 belong to A.

Revision Table: Key Concepts

Additional Information: Properties of Cartesian Products

• The order of elements in an ordered pair matters, i.e., (a, b) is generally not equal to (b, a) unless a = b.
• If A or B is an empty set (\(\emptyset\)), then A × B is also an empty set (\(\emptyset\)).
• If A and B are finite sets, then \(|\)A × B\(|\) = \(|\)A\(|\) \(\times\) \(|\)B\(|\).
• The Cartesian product is not commutative, i.e., A × B \(\neq\) B × A unless A = B or A or B is empty. In our case, A × A = A × A is trivially true because it's the same product.
• The Cartesian product can be extended to more than two sets, e.g., A × B × C would be a set of ordered triples (a, b, c).