We are given the set $S = \{1, 2, 3, 4, 5, 6\}$. We need to find the cardinality (number of elements) of the Cartesian product $S \times S$. The Cartesian product $S \times S$ is the set of all ordered pairs $(a, b)$ where $a \in S$ and $b \in S$. We can find $|S \times S|$ by listing all its elements, or more easily, by calculating $|S| \times |S|$.

1. Problem Description

We are given the set $S = \{1, 2, 3, 4, 5, 6\}$. We need to find the cardinality (number of elements) of the Cartesian product $S \times S$. The Cartesian product $S \times S$ is the set of all ordered pairs $(a, b)$ where $a \in S$ and $b \in S$. We can find $|S \times S|$ by listing all its elements, or more easily, by calculating $|S| \times |S|$.

2. Solution Steps

First, we find the cardinality of the set $S$. Since $S = \{1, 2, 3, 4, 5, 6\}$, the number of elements in $S$ is

6. So, $|S| = 6$.

The Cartesian product $S \times S$ consists of ordered pairs $(a, b)$ where $a \in S$ and $b \in S$. The number of elements in the Cartesian product $S \times S$ is given by

$|S \times S| = |S| \times |S|$.

Substituting $|S| = 6$, we have

$|S \times S| = 6 \times 6 = 36$.

Listing the elements of $S \times S$ is possible but tedious. Here is the beginning:

$S \times S = \{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), \dots (6,6) \}$. There are 6 rows and 6 columns, thus a total of $6 \times 6 = 36$ elements.

3. Final Answer

$|S \times S| = 36$