The Relation R R R Is Defined By The Ordered Pairs Listed Below.$R = {(-7, -3), (-1, 0), (0, 18), (1, 14), (7, 10)}$1. The Domain Of R R R Is ${\ \square\ }$2. The Range Of R R R Is ${\ \square\

Introduction

In mathematics, a relation is a set of ordered pairs that define a connection between two or more elements. The relation $R$ is defined by the ordered pairs listed below: $R = \{(-7, -3), (-1, 0), (0, 18), (1, 14), (7, 10)\}$. In this article, we will explore the concept of domain and range in the context of relation $R$.

Domain of R

The domain of a relation is the set of all first elements in the ordered pairs. In other words, it is the set of all values that appear as the first element in the ordered pairs. To find the domain of $R$, we need to identify the first element in each ordered pair.

• $(-7, -3)$: The first element is $-7$.
• $(-1, 0)$: The first element is $-1$.
• $(0, 18)$: The first element is $0$.
• $(1, 14)$: The first element is $1$.
• $(7, 10)$: The first element is $7$.

The domain of $R$ is the set of all these first elements, which is $\{-7, -1, 0, 1, 7\}$. Therefore, the domain of $R$ is $\{\boxed{-7, -1, 0, 1, 7}\}$.

Range of R

The range of a relation is the set of all second elements in the ordered pairs. In other words, it is the set of all values that appear as the second element in the ordered pairs. To find the range of $R$, we need to identify the second element in each ordered pair.

• $(-7, -3)$: The second element is $-3$.
• $(-1, 0)$: The second element is $0$.
• $(0, 18)$: The second element is $18$.
• $(1, 14)$: The second element is $14$.
• $(7, 10)$: The second element is $10$.

The range of $R$ is the set of all these second elements, which is $\{-3, 0, 18, 14, 10\}$. Therefore, the range of $R$ is $\{\boxed{-3, 0, 18, 14, 10}\}$.

Conclusion

In conclusion, the domain of $R$ is the set of all first elements in the ordered pairs, which is $\{-7, -1, 0, 1, 7\}$. The range of $R$ is the set of all second elements in the ordered pairs, which is $\{-3, 0, 18, 14, 10\}$. Understanding the domain and range of a relation is essential in mathematics, as it helps us to analyze and interpret the relationships between different elements.

Key Takeaways

• The domain of a relation is the set of all first elements in the ordered pairs.
• The range of a relation is the set of all second elements in the ordered pairs.
• Understanding the domain and range of a relation is essential in mathematics.

Frequently Asked Questions

Q: What is the domain of R?

A: The domain of R is the set of all first elements in the ordered pairs, which is $\{-7, -1, 0, 1, 7\}$.

Q: What is the range of R?

A: The range of R is the set of all second elements in the ordered pairs, which is $\{-3, 0, 18, 14, 10\}$.

Q: Why is understanding the domain and range of a relation important?

Q&A: Frequently Asked Questions

Q: What is the domain of R?

A: The domain of R is the set of all first elements in the ordered pairs, which is $\{-7, -1, 0, 1, 7\}$.

Q: What is the range of R?

A: The range of R is the set of all second elements in the ordered pairs, which is $\{-3, 0, 18, 14, 10\}$.

Q: Why is understanding the domain and range of a relation important?

A: Understanding the domain and range of a relation is essential in mathematics, as it helps us to analyze and interpret the relationships between different elements.

Q: Can you give an example of a relation with a domain and range?

A: Consider the relation $S = \{(2, 4), (3, 6), (4, 8), (5, 10)\}$. The domain of S is the set of all first elements in the ordered pairs, which is $\{2, 3, 4, 5\}$. The range of S is the set of all second elements in the ordered pairs, which is $\{4, 6, 8, 10\}$.

Q: How do you find the domain and range of a relation?

A: To find the domain and range of a relation, you need to identify the first and second elements in each ordered pair. The domain is the set of all first elements, and the range is the set of all second elements.

Q: Can the domain and range of a relation be the same?

A: Yes, the domain and range of a relation can be the same. For example, consider the relation $T = \{(1, 1), (2, 2), (3, 3), (4, 4)\}$. The domain of T is the set of all first elements, which is $\{1, 2, 3, 4\}$. The range of T is also the set of all second elements, which is $\{1, 2, 3, 4\}$. In this case, the domain and range of T are the same.

Q: What is the difference between the domain and range of a relation?

A: The domain of a relation is the set of all first elements in the ordered pairs, while the range of a relation is the set of all second elements in the ordered pairs. In other words, the domain is the set of all values that appear as the first element in the ordered pairs, and the range is the set of all values that appear as the second element in the ordered pairs.

Q: Can the domain and range of a relation be empty?

A: Yes, the domain and range of a relation can be empty. For example, consider the relation $U = \{(\emptyset, \emptyset)\}$. The domain of U is the set of all first elements, which is $\{\emptyset\}$. The range of U is also the set of all second elements, which is $\{\emptyset\}$. In this case, the domain and range of U are not empty.

Q: What is the significance of the domain and range of a relation in real-life scenarios?

A: The domain and range of a relation are significant in real-life scenarios because they help us to understand the relationships between different elements. For example, in a database, the domain of a relation represents the set of all possible values that can be stored in a particular column, while the range of a relation represents the set of all possible values that can be retrieved from a particular column.

Q: Can the domain and range of a relation be infinite?

A: Yes, the domain and range of a relation can be infinite. For example, consider the relation $V = \{(n, 2n) | n \in \mathbb{N}\}$. The domain of V is the set of all natural numbers, which is infinite. The range of V is also the set of all even numbers, which is infinite. In this case, the domain and range of V are infinite.