The image you uploaded shows Exercise 2.5.9, which involves Cartesian products of sets. Here are the questions extracted from the image:

Exercise 2.5.9

Let $A, C \subseteq X$ and $B, D \subseteq Y$.

• (a) True or false? $A \times B \subseteq C \times D$ if and only if $A \subseteq C$ and $B \subseteq D$.
• (b) True or false? $(A \cap C) \times (B \cap D) = (A \times B) \cap (C \times D)$.
• (c) True or false? $(A \times C) \cup (B \times D) = (A \cup B) \times (C \cup D)$.

Analysis and Solutions:

1. Part (a): The question asks if $A \times B \subseteq C \times D$ is equivalent to saying $A \subseteq C$ and $B \subseteq D$.

   • True. This is a known property of Cartesian products. If $A \times B \subseteq C \times D$, it means every pair in $A \times B$ also exists in $C \times D$, implying that $A \subseteq C$ and $B \subseteq D$.

2. Part (b): This asks whether the Cartesian product of the intersections equals the intersection of the Cartesian products.

   • True. This follows from the distributive property of intersections over Cartesian products. Specifically: $(A \cap C) \times (B \cap D) = \{(a, b) : a \in A \cap C \text{ and } b \in B \cap D\}$ is equal to: $(A \times B) \cap (C \times D) = \{(a, b) : (a \in A \text{ and } b \in B) \text{ and } (a \in C \text{ and } b \in D)\}$ Both result in the same set.

3. Part (c): The question asks if the union of two Cartesian products is equal to the Cartesian product of the unions.

   • False. The union of Cartesian products does not distribute in the same way. For example, if $A = \{1\}, C = \{2\}, B = \{3\}, D = \{4\}$, then $(A \times C) \cup (B \times D)$ would be $\{(1,2), (3,4)\}$, which is not the same as $(A \cup B) \times (C \cup D) = \{(1,2), (1,4), (3,2), (3,4)\}$.

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Related Questions:

1. Can you prove why $A \times B \subseteq C \times D$ implies $A \subseteq C$ and $B \subseteq D$?
2. How does the Cartesian product behave under other set operations like difference or symmetric difference?
3. Can the distributive property of Cartesian products be extended to more complex set combinations?
4. Why does the Cartesian product not distribute over unions as it does over intersections?
5. How does the Cartesian product interact with power sets and their relationships?

Tip:

Remember, the Cartesian product of two sets involves pairing elements from both sets, which leads to different behaviors depending on whether you are working with intersections or unions!