https://r2.erweima.ai/mathbot/image/172673760298817g6y5oe.jpg

Let's solve the tasks from the image one by one.

### 1. Given two sets \( A = \{a, b\} \) and \( B = \{c, d\} \):

#### a. Identify the elements in \( A \times B \) (the Cartesian product of \( A \) and \( B \)).
The Cartesian product \( A \times B \) is the set of all ordered pairs \( (x, y) \) where \( x \in A \) and \( y \in B \).

\[
A \times B = \{(a, c), (a, d), (b, c), (b, d)\}
\]

#### b. List all the possible relations that can be made in \( A \times B \).
A relation from \( A \) to \( B \) is any subset of \( A \times B \). The number of possible relations is the number of subsets of \( A \times B \), which is \( 2^{|A \times B|} \).

Since \( A \times B \) contains 4 elements, the total number of subsets (relations) is \( 2^4 = 16 \).

The possible relations (subsets) include:
1. \( \varnothing \) (the empty relation)
2. \( \{(a, c)\} \)
3. \( \{(a, d)\} \)
4. \( \{(b, c)\} \)
5. \( \{(b, d)\} \)
6. \( \{(a, c), (a, d)\} \)
7. \( \{(a, c), (b, c)\} \)
8. \( \{(a, c), (b, d)\} \)
9. \( \{(a, d), (b, c)\} \)
10. \( \{(a, d), (b, d)\} \)
11. \( \{(b, c), (b, d)\} \)
12. \( \{(a, c), (a, d), (b, c)\} \)
13. \( \{(a, c), (a, d), (b, d)\} \)
14. \( \{(a, c), (b, c), (b, d)\} \)
15. \( \{(a, d), (b, c), (b, d)\} \)
16. \( A \times B \) itself, \( \{(a, c), (a, d), (b, c), (b, d)\} \)

#### c. Which of the relations are functions?
A relation from \( A \) to \( B \) is a function if each element of \( A \) is related to exactly one element of \( B \). So, for a subset of \( A \times B \) to be a function, for each \( x \in A \), there should be exactly one pair \( (x, y) \in A \times B \).

Valid functions include:
1. \( \{(a, c), (b, c)\} \)
2. \( \{(a, d), (b, c)\} \)
3. \( \{(a, c), (b, d)\} \)
4. \( \{(a, d), (b, d)\} \)

These are the only relations where each element of \( A \) is associated with exactly one element in \( B \).

---

### 2. Let \( A = \{5, 6, 2, 8, 9, 10\} \) and \( B = \{7, 8, 9, 10, 11, 12\} \):

#### a. Identify the elements in \( A \times B \).
The Cartesian product \( A \times B \) contains all possible pairs \( (x, y) \), where \( x \in A \) and \( y \in B \). Therefore,

\[
A \times B = \{(5, 7), (5, 8), (5, 9), (5, 10), (5, 11), (5, 12), \dots, (10, 7), (10, 8), (10, 9), (10, 10), (10, 11), (10, 12)\}
\]

This results in 36 pairs.

#### b. Define a relation \( R \) from \( A \) to \( B \) by:
\[
R = \{(x, y) \ | \ y - x = 2 \}
\]
To find the pairs \( (x, y) \) such that \( y - x = 2 \), we calculate:
\[
(5, 7), (6, 8), (8, 10), (9, 11), (10, 12)
\]

Thus, the relation \( R \) is:
\[
R = \{(5, 7), (6, 8), (8, 10), (9, 11), (10, 12)\}
\]

#### c. Is the relation \( R \) a function?
Yes, the relation \( R \) is a function because each element of \( A \) is related to exactly one element of \( B \). Each \( x \in A \) has exactly one corresponding \( y \in B \) satisfying \( y - x = 2 \).

---

Would you like more details or explanations on any part? Here are some related questions:

1. What is the significance of Cartesian products in set theory?
2. How can we determine the number of functions from one set to another?
3. What is the difference between a relation and a function?
4. Can relations be represented graphically? How?
5. What are real-life examples of Cartesian products and functions?

**Tip:** A relation can be a function only if no two ordered pairs have the same first element associated with different second elements.

Perform the relations in two sets:
1. Given two sets A = {a, b} and B = {c, d}:
  a. Identify the elements in A x B.
  b. List all the possible relations that can be made in A x B.
  c. Which of the relations are functions?

2. Let A = {5, 6, 2, 8, 9, 10} and B = {7, 8, 9, 10, 11, 12}:
  a. Identify the elements in A x B.
  b. Define a relation R from A to B by R = {(x, y) | y - x = 2}.
  c. Is the relation a function?

Learn how to perform relations and functions between two sets with detailed examples. This problem covers key concepts like Cartesian product, relations, and functions. Suitable for students in Grades 8-10.

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