Let’s go through the parts one by one, using set-roster notation to solve them.

Given Sets:

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

(a) $A \times (B \cup C)$

First, we compute the union of sets $B$ and $C$: $B \cup C = \{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$

Now, we take the Cartesian product of $A$ and $B \cup C$: $A \times (B \cup C) = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$

(b) $(A \times B) \cup (A \times C)$

First, compute the Cartesian products separately: $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$ $A \times C = \{(a, 2), (a, 3), (b, 2), (b, 3)\}$

Now, take the union of these two sets: $(A \times B) \cup (A \times C) = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$

(c) $A \times (B \cap C)$

First, compute the intersection of sets $B$ and $C$: $B \cap C = \{2\}$

Now, take the Cartesian product of $A$ and $B \cap C$: $A \times (B \cap C) = \{(a, 2), (b, 2)\}$

(d) $(A \times B) \cap (A \times C)$

Now, find the intersection of the two Cartesian products: $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$ $A \times C = \{(a, 2), (a, 3), (b, 2), (b, 3)\}$

The common elements in both sets are: $(A \times B) \cap (A \times C) = \{(a, 2), (b, 2)\}$

Final Results:

• (a) $A \times (B \cup C) = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$
• (b) $(A \times B) \cup (A \times C) = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$
• (c) $A \times (B \cap C) = \{(a, 2), (b, 2)\}$
• (d) $(A \times B) \cap (A \times C) = \{(a, 2), (b, 2)\}$

Would you like more details or have any questions on this? Here are some related questions:

1. What are the properties of the Cartesian product of two sets?
2. How do union and intersection affect the size of sets in set theory?
3. Can a Cartesian product result in an empty set?
4. What is the significance of the Cartesian product in relational databases?
5. How does set theory relate to Venn diagrams?

Tip: The Cartesian product of two sets can significantly increase the number of elements when both sets are large.