To solve the problem "Find $A \times B$", where $A = \{a, b\}$ and $B = \{1, 2, 3\}$, we are asked to find the Cartesian product of two sets.

Cartesian Product Definition:

For two sets $A$ and $B$, the Cartesian product $A \times B$ is defined as the set of all ordered pairs $(x, y)$, where $x \in A$ and $y \in B$.

Step-by-step Calculation:

1. $A = \{a, b\}$
2. $B = \{1, 2, 3\}$

The Cartesian product $A \times B$ will include all pairs where the first element is from $A$ and the second is from $B$. So, we pair each element of $A$ with each element of $B$:

• Pair $a$ with all elements of $B$:

  • $(a, 1)$
  • $(a, 2)$
  • $(a, 3)$

• Pair $b$ with all elements of $B$:

  • $(b, 1)$
  • $(b, 2)$
  • $(b, 3)$

Thus, $A \times B = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$.

Matching the Answer Choices:

Looking at the provided options in the image:

• The correct answer is the third option: $\{(a, 1), (b, 1), (a, 2), (b, 2), (a, 3), (b, 3)\}$

Would you like more details on Cartesian products or any other related topics?

Here are 5 follow-up questions:

1. How does the Cartesian product change if $B$ had more or fewer elements?
2. Can the Cartesian product be applied to three or more sets?
3. What are some applications of Cartesian products in computer science or data analysis?
4. How is the Cartesian product related to the concept of a "relation" in set theory?
5. What is the Cartesian product of a set with itself?

Tip: The size of the Cartesian product $A \times B$ is the product of the sizes of $A$ and $B$.