The image contains a mathematical note on the Cartesian product of sets. Here's a breakdown of what it contains:

1. Definition:

   • The Cartesian product of two sets $A$ and $B$, denoted as $A \times B$, is the set of all ordered pairs $(a, b)$, where $a \in A$ and $b \in B$.
   • Mathematically, it's written as: $A \times B = \{(a, b) : a \in A, b \in B\}$

2. Example Sets:

   • $A = \{0, 3\}$
   • $B = \{a, b, c\}$

3. Tasks:

   • You are asked to find:
     1. $A \times B$
     2. $A \times A$
     3. $B \times A$
     4. $B \times B$

Solutions:

1. $A \times B$:

   • The ordered pairs from $A$ and $B$ are: $A \times B = \{(0, a), (0, b), (0, c), (3, a), (3, b), (3, c)\}$

2. $A \times A$:

   • The ordered pairs from $A$ and $A$ are: $A \times A = \{(0, 0), (0, 3), (3, 0), (3, 3)\}$

3. $B \times A$:

   • The ordered pairs from $B$ and $A$ are: $B \times A = \{(a, 0), (a, 3), (b, 0), (b, 3), (c, 0), (c, 3)\}$

4. $B \times B$:

   • The ordered pairs from $B$ and $B$ are: $B \times B = \{(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)\}$

Would you like more details or have any questions?

5 Related Questions:

1. What happens if one of the sets is empty in a Cartesian product?
2. How does the Cartesian product relate to functions?
3. Can we extend the Cartesian product to more than two sets?
4. How does the Cartesian product behave when the sets are infinite?
5. What is the relationship between Cartesian products and coordinate systems?

Tip:

The size of the Cartesian product $A \times B$ is the product of the cardinalities of $A$ and $B$, i.e., $|A \times B| = |A| \times |B|$.