The Cartesian product A×A is defined. Find the sum of all the elements 

To find the sum of all the elements of the Cartesian product \( A \times A \), we first need to understand the problem and the definition of the Cartesian product in this context.

The Cartesian product \( A \times A \) is the set of all ordered pairs \( (a, b) \) where both \( a \) and \( b \) are elements of set \( A \).

Let's denote the set \( A \) as \( A = \{a_1, a_2, \ldots, a_n\} \).

The Cartesian product \( A \times A \) consists of all pairs \( (a_i, a_j) \) for \( i, j \) in \( \{1, 2, \ldots, n\} \). Each element in the Cartesian product is an ordered pair, and we need to find the sum of all these pairs.

For clarity, let's consider an example. Suppose \( A = \{1, 2\} \).

Then the Cartesian product \( A \times A \) is:
\[ A \times A = \{(1, 1), (1, 2), (2, 1), (2, 2)\} \]

To find the sum of all elements in this Cartesian product, we sum the elements of each pair:
\[ (1, 1) \rightarrow 1 + 1 = 2 \]
\[ (1, 2) \rightarrow 1 + 2 = 3 \]
\[ (2, 1) \rightarrow 2 + 1 = 3 \]
\[ (2, 2) \rightarrow 2 + 2 = 4 \]

So, the total sum is:
\[ 2 + 3 + 3 + 4 = 12 \]

Now, let's generalize this. If \( A = \{a_1, a_2, \ldots, a_n\} \), the Cartesian product \( A \times A \) contains \( n \times n = n^2 \) pairs. Each element \( a_i \) appears \( n \) times as the first element and \( n \) times as the second element in these pairs.

Thus, the total sum of all pairs in \( A \times A \) is:
\[ \text{Sum} = n \cdot \sum_{i=1}^{n} a_i + n \cdot \sum_{j=1}^{n} a_j = 2n \cdot \sum_{k=1}^{n} a_k \]

If we denote the sum of all elements in set \( A \) as \( S \), where \( S = \sum_{k=1}^{n} a_k \), then the total sum of all elements in the Cartesian product \( A \times A \) is:
\[ \text{Total sum} = 2n \cdot S \]

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. What is the Cartesian product of a set with itself called in terms of set operations?
2. How does the sum of the elements change if the set \( A \) contains negative numbers?
3. What is the Cartesian product \( A \times A \) if \( A = \{0, 1, 2\} \)?
4. How can we interpret the sum of all elements of \( A \times A \) geometrically?
5. What is the Cartesian product \( A \times B \) if \( A = \{1, 3\} \) and \( B = \{2, 4\} \)?
6. If \( A = \{a, b\} \), what is the sum of all elements in \( A \times A \) in terms of \( a \) and \( b \)?
7. How would you explain the Cartesian product in terms of matrix operations?
8. What are the properties of the Cartesian product related to commutativity and associativity?

**Tip:** When calculating the sum of all elements in a Cartesian product, always consider the frequency of each element's occurrence in the product pairs. This approach helps simplify the calculation.

Discover how to find the sum of all elements in the Cartesian product A×A using set theory principles. This problem explores the definition of Cartesian product, set operations, and mathematical calculation techniques. Suitable for advanced high school students and beyond.

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