Let’s analyze each question step-by-step.

Question 1: Cartesian Product Statements

We are given three statements, and we need to determine which are correct.

1. Statement 1: $P = \{m, n\}$ and $Q = \{n, m\}$, then $P \times Q = \{(m, n), (n, m)\}$.

   • For $P \times Q$, we take every possible ordered pair $(a, b)$ where $a \in P$ and $b \in Q$. Since $P = Q = \{m, n\}$, the Cartesian product $P \times Q$ should yield $\{(m, m), (m, n), (n, m), (n, n)\}$.
   • This statement is incorrect because it only includes two pairs instead of four.

2. Statement 2: If $A$ and $B$ are non-empty sets, then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$.

   • This is the definition of the Cartesian product for non-empty sets. Thus, this statement is correct.

3. Statement 3: If $A = \{1, 2\}$, $B = \{3, 4\}$, then $A \times (B \cap \emptyset) = \emptyset$.

   • Since $B \cap \emptyset = \emptyset$, $A \times \emptyset = \emptyset$ because there are no elements in $\emptyset$ to pair with elements of $A$.
   • This statement is correct.

So, the correct statements are 2 and 3 only.

Answer for Question 1: 2 and 3 only

Question 2: Set Expressions in Increasing Order of Size

Let’s go through the expressions provided:

• $|P - Q|$: Represents the number of elements in $P$ that are not in $Q$.
• $|P \oplus Q|$: Represents the symmetric difference, which is the set of elements in $P$ or $Q$ but not in both.
• $|P| + |Q|$: Sum of the cardinalities of $P$ and $Q$.
• $|P \cup Q|$: Represents the union, which is all elements in either $P$ or $Q$ without duplicates.

We need to determine the correct order of these expressions in terms of increasing size.

1. General Order:
   • Typically, $|P - Q| \leq |P \oplus Q| \leq |P \cup Q|$.
   • $|P| + |Q|$ can be larger than $|P \cup Q|$ because it double-counts the intersection of $P$ and $Q$.

Analyzing the answer choices:

• Choice A: $|P - Q| < |P \oplus Q| < |P| + |Q| < |P \cup Q|$
• Choice B: $|P - Q| < |P \oplus Q| < |P \cup Q| < |P| + |Q|$
• Choice C: $|P \oplus Q| < |P - Q| < |P| + |Q| < |P \cup Q|$
• Choice D: $|P \oplus Q| < |P| + |Q| < |P - Q| < |P \cup Q|$

Based on typical properties of set operations, Choice B seems logically ordered, as the union $|P \cup Q|$ would likely be less than the sum $|P| + |Q|$.

Answer for Question 2: B

Let me know if you'd like further details on any part of these solutions or have additional questions.

Here are five related questions to deepen understanding:

1. What is the Cartesian product and how does it differ when sets have overlapping elements?
2. Why is the symmetric difference used, and how does it relate to the union and intersection of sets?
3. How do the properties of empty sets affect operations like Cartesian products and intersections?
4. What is the difference between $|P \cup Q|$ and $|P| + |Q|$ in terms of counting elements?
5. How do set operations behave when applied to finite versus infinite sets?

Tip: Always visualize sets as Venn diagrams for complex set operations; it helps clarify overlapping areas and relationships between operations.