The question asks to select the false statement, given the sets:

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

Let’s analyze each option:

Option A: $|A \times B| = 5$

The Cartesian product $A \times B$ represents all possible pairs $(x, y)$ where $x \in A$ and $y \in B$. Since $A$ has 2 elements and $B$ has 3 elements, the total number of pairs is:

$|A \times B| = 2 \times 3 = 6$

This is false since the statement says $|A \times B| = 5$.

Option B: $(b, 3) \in A \times B$

The pair $(b, 3)$ is indeed a valid pair in $A \times B$ because $b \in A$ and $3 \in B$. Therefore, this statement is true.

Option C: $A \cap A^2 = \emptyset$

The notation $A^2$ typically refers to the Cartesian product $A \times A$, i.e., the set of ordered pairs $(x, y)$ where both $x$ and $y$ are elements of $A$. So, $A^2 = \{(a, a), (a, b), (b, a), (b, b)\}$.

The intersection $A \cap A^2$ compares the set $A$ (which consists of elements $a$ and $b$) with the set of pairs in $A^2$. Since no single elements exist in $A^2$, this intersection is indeed the empty set.

Thus, this statement is true.

Option D: $(b, a) \in A^2$

The pair $(b, a)$ is a valid element of $A^2$ because both $b$ and $a$ are in $A$. Therefore, this statement is true.

Conclusion:

The false statement is Option A: $|A \times B| = 5$.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. What is the general formula for the size of the Cartesian product of two sets?
2. Can you explain the meaning of intersections between sets and Cartesian products?
3. How does one interpret the notation $A^2$ in set theory?
4. What are the properties of Cartesian products with empty sets?
5. How can we visually represent the Cartesian product of two sets?

Tip: Remember that the Cartesian product always generates ordered pairs, and the number of pairs is the product of the cardinalities of the individual sets.