{(1,2), (-2,3), (1,3), (-2,2)} is what type of relation?

A. one-to-many
B. many-to-one
C. one-to-one
D. many-to-many

To determine the type of relation represented by the set \(\{(1,2), (-2,3), (1,3), (-2,2)\}\), we need to examine how elements from the domain (the first components of the pairs) relate to elements in the codomain (the second components of the pairs).

### Analysis of the Relation:

1. **Domain**: The set of all first components of the pairs: \(\{1, -2\}\).
2. **Codomain**: The set of all second components of the pairs: \(\{2, 3\}\).

#### Pair Analysis:
- **(1, 2)**: The element \(1\) in the domain is associated with \(2\) in the codomain.
- **(-2, 3)**: The element \(-2\) in the domain is associated with \(3\) in the codomain.
- **(1, 3)**: The element \(1\) in the domain is also associated with \(3\) in the codomain.
- **(-2, 2)**: The element \(-2\) in the domain is also associated with \(2\) in the codomain.

### Determination of the Type:

- **One-to-One**: Each element in the domain is paired with a unique element in the codomain. This is not the case here because \(1\) is paired with both \(2\) and \(3\), and \(-2\) is paired with both \(3\) and \(2\).
  
- **One-to-Many**: An element in the domain is related to multiple elements in the codomain. This is applicable here because:
  - The element \(1\) in the domain is related to both \(2\) and \(3\).
  - The element \(-2\) in the domain is related to both \(3\) and \(2\).

- **Many-to-One**: Multiple elements in the domain are paired with the same element in the codomain. This is not applicable here because there are no elements in the codomain with more than one element from the domain.

- **Many-to-Many**: Multiple elements in the domain are related to multiple elements in the codomain, and vice versa. This is also applicable here, as each element in the domain can be paired with multiple elements in the codomain, and each element in the codomain is associated with multiple elements from the domain.

### Conclusion:
The relation \(\{(1,2), (-2,3), (1,3), (-2,2)\}\) is a **many-to-many** relation.

Therefore, the correct answer is:

**D. many-to-many**

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

1. What are the characteristics of a one-to-many relation?
2. How does a many-to-one relation differ from one-to-many?
3. Can you provide an example of a one-to-one relation?
4. What is the significance of the domain and codomain in a relation?
5. How can we visually represent a relation using a graph?
6. What is the difference between a function and a relation?
7. Can a relation be both one-to-many and many-to-one at the same time?
8. How can we identify the type of relation from a set of ordered pairs?

**Tip:** To identify the type of relation, focus on how elements in the domain map to elements in the codomain. This helps to quickly determine whether the relation is one-to-one, one-to-many, many-to-one, or many-to-many.

Learn how to identify the type of relation from a set of ordered pairs. Understand concepts such as one-to-many, many-to-one, one-to-one, and many-to-many relationships through detailed examples. Suitable for students and learners interested in set theory and relations in mathematics.

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