The set $A = \{1, 1\}$ actually contains only one distinct element, which is $1$. In set theory, duplicate elements are not counted, so this set simplifies to:

$A = \{1\}$

Now, if we take the Cartesian product of $A$ with itself three times, i.e., $A \times A \times A$, we are essentially finding the set of all ordered triplets $(a, b, c)$ where each of $a$, $b$, and $c$ is an element of $A$.

Since $A = \{1\}$, every element in $A \times A \times A$ will be the triplet $(1, 1, 1)$. Therefore,

$A \times A \times A = \{(1, 1, 1)\}$

Summary:

The Cartesian product $A \times A \times A$ is simply $\{(1, 1, 1)\}$.

Would you like further explanation or clarification? Here are 5 related questions you might find helpful:

1. What is the Cartesian product of two sets, $A \times B$?
2. How do you find the number of elements in a Cartesian product of sets?
3. Can a set contain duplicate elements, and how does this affect its size?
4. How do you find the Cartesian product of an empty set with any set?
5. What happens when you take the Cartesian product of a set with itself multiple times?

Tip: The Cartesian product of two sets results in a set of all possible ordered pairs formed by taking one element from each set.