The problem states that we have subsets $A_1, A_2, A_3$ of the set $\{1, 2, 3, 4, 5\}$, and they need to satisfy the condition:

$(A_1 \cup A_2) \cap (A_2 \cup A_3) = \{1, 2, 3, 4, 5\}.$

This means that every element in the set $\{1, 2, 3, 4, 5\}$ must belong to both $A_1 \cup A_2$ and $A_2 \cup A_3$.

To satisfy this condition, each element in $\{1, 2, 3, 4, 5\}$ must appear in either:

• $A_1$ or $A_2$, and
• $A_2$ or $A_3$.

Analysis for Each Element

For any element $x \in \{1, 2, 3, 4, 5\}$:

1. It can belong to both $A_1$ and $A_3$ independently.
2. It must be in at least one of $A_1$ or $A_2$ and at least one of $A_2$ or $A_3$.

This results in the following possible placements for each element $x$:

1. $x \in A_1$ and $x \in A_3$.
2. $x \in A_1$ and $x \notin A_3$.
3. $x \notin A_1$ and $x \in A_3$.
4. $x \in A_2$ only (if $x$ is not in either $A_1$ or $A_3$).

Each element has 4 possible configurations to satisfy the condition.

Total Configurations

Since there are 5 elements in $\{1, 2, 3, 4, 5\}$, and each element can be assigned to $A_1$, $A_2$, and $A_3$ in 4 ways to satisfy the condition, the total number of ways to assign elements is:

$N = 4^5 = 1024.$

Finding the Last Two Digits of $N$

To find the last two digits of $1024$, we calculate $1024 \mod 100$:

$1024 \mod 100 = 24.$

Answer

The last two digits of $N$ are $\boxed{24}$.

Would you like more details on any part of the solution, or do you have further questions?

Here are some related questions that expand on this concept:

1. What if the set had more elements, say $\{1, 2, \dots, 6\}$? How would the answer change?
2. How would the solution be affected if we required every element to appear in $A_1$ or $A_3$ but not necessarily in $A_2$?
3. What if we changed the condition to $(A_1 \cap A_2) \cup (A_2 \cap A_3) = \{1, 2, 3, 4, 5\}$?
4. Can you generalize this approach to subsets $A_1, A_2, \ldots, A_n$?
5. How would the answer differ if we were only asked for the number of ways to create each subset without regard to the specific ordering of $A_1, A_2,$ and $A_3$?

Tip: When analyzing sets with union and intersection conditions, consider each element independently and account for the constraints it must satisfy across the sets.