To determine which of the given relations contains the maximum number of elements, we need to calculate the Cartesian product of each pair of sets. The Cartesian product $A \times B$ is defined as the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

Let's calculate each one:

1. $A \times B$:

   • $A = \{1, 2, 3\}$
   • $B = \{5, 6\}$
   • Number of elements in $A \times B$ = $|A| \times |B| = 3 \times 2 = 6$

2. $B \times A$:

   • $B = \{5, 6\}$
   • $A = \{1, 2, 3\}$
   • Number of elements in $B \times A$ = $|B| \times |A| = 2 \times 3 = 6$

3. $B \times B$:

   • $B = \{5, 6\}$
   • Number of elements in $B \times B$ = $|B| \times |B| = 2 \times 2 = 4$

4. $A \times A$:

   • $A = \{1, 2, 3\}$
   • Number of elements in $A \times A$ = $|A| \times |A| = 3 \times 3 = 9$

Comparing the number of elements in each relation:

• $A \times B$: 6 elements
• $B \times A$: 6 elements
• $B \times B$: 4 elements
• $A \times A$: 9 elements

Thus, the relation with the maximum number of elements is $A \times A$, which contains 9 elements.

Answer:

(d) $A \times A$

Would you like details on how these calculations are performed or have any other questions?

Here are some related questions you might find interesting:

1. What is the Cartesian product of two sets?
2. How do you determine the number of elements in a Cartesian product?
3. Can the Cartesian product be performed on more than two sets?
4. What are some applications of Cartesian products in real-world problems?
5. How can you visualize Cartesian products?
6. What happens if one or both sets in a Cartesian product are empty?
7. What are the properties of Cartesian products?
8. How does the concept of Cartesian products relate to other mathematical operations, such as unions and intersections?

Tip: When dealing with Cartesian products, always remember that the number of elements in the product set is the product of the number of elements in each individual set.