Let 

*A* = {*a*, *b*},
 
*B* = {1, 2},

 and 

*C* = {2, 3}.

 Use set-roster notation to write each of the following sets.

(a)

*A* ✕ (*B* ∪ *C*) = 
 
(b)

(*A* ✕ *B*) ∪ (*A* ✕ *C*) = 
 
(c)

*A* ✕ (*B* ∩ *C*) = 
 
(d)

(*A* ✕ *B*) ∩ (*A* ✕ *C*) =

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.

Let A = {a, b}, B = {1, 2}, and C = {2, 3}. Use set-roster notation to write each of the following sets. (a) A ✕ (B ∪ C) = (b) (A ✕ B) ∪ (A ✕ C) = (c) A ✕ (B ∩ C) = (d) (A ✕ B) ∩ (A ✕ C) =

Explore the concepts of set theory with a detailed solution to finding Cartesian products, unions, and intersections. Learn how to apply set-roster notation to sets A = {a, b}, B = {1, 2}, and C = {2, 3}. Suitable for students in Grades 9-12.

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