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.