How can I calculate $A\times A$ in a cartesian product? What is the Cartesian product of a set with itself?  For example $A=\{1,2,3\}$. $A\times A=$?
 A: It doesn’t matter what the sets are, $A\times B$ is always the set of all ordered pairs $\langle x,y\rangle$ such that $x\in A$ and $y\in B$. This is true whether or not $B=A$. In your case we have $\{1,2,3\}\times\{1,2,3\}$, which is every ordered pair $\langle x,y\rangle$ such that $x\in\{1,2,3\}$ and $y\in\{1,2,3\}$. In other words, it’s every ordered pair whose components are both in the set $A=\{1,2,3\}$. There are $3^2=9$ of them. In a small example like this you can easily systematically list them in tabular form:
$$\begin{array}{c|ccc}
&1&2&3\\ \hline
1&\langle 1,1\rangle&\langle 1,2\rangle&\langle 1,3\rangle\\
2&\langle 2,1\rangle&\langle 2,2\rangle&\langle 2,3\rangle\\
3&\langle 3,1\rangle&\langle 3,2\rangle&\langle 3,3\rangle\\
\end{array}$$
Now just write out the set: $A\times A=\{\langle 1,1\rangle,\dots \text{What goes here?}\}$.
A: No differently than you'd compute the Cartesian product of two differing sets. Recall that $A\times B$ is the set of all ordered pairs $(a,b)$ such that $a \in A$ and $b \in B$. In the case where $A = B = \left\{1,2,3\right\}$:
$$\left\{(1,1),(1,2)(1,3),(2,1),(2,2),(2,3),(3,1),(3,2)(3,3)\right\}$$
