The Relation R R R Is Defined By The Ordered Pairs Listed Below:$R={(-9,20),(-3,10),(0,13),(2,-10),(8,-3)}$1. The Domain Of R R R Is ${ \square }$2. The Range Of R R R Is ${ \square }$3. Is

Introduction

In mathematics, a relation is a set of ordered pairs that define a connection between two sets. The relation $R$ is defined by the ordered pairs listed below:

$R=\{(-9,20),(-3,10),(0,13),(2,-10),(8,-3)\}$

In this article, we will analyze the relation $R$ and determine its domain, range, and other properties.

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.

• Definition of Domain: The domain of a relation $R$ is the set of all first elements in the ordered pairs, denoted as $D(R)$.

• Domain of R: To find the domain of $R$, we need to identify the first element in each ordered pair.

  • $(-9,20)$: The first element is $-9$.
  • $(-3,10)$: The first element is $-3$.
  • $(0,13)$: The first element is $0$.
  • $(2,-10)$: The first element is $2$.
  • $(8,-3)$: The first element is $8$.

• Domain of R: The domain of $R$ is the set of all these first elements, which is $\{-9,-3,0,2,8\}$.

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.

• Definition of Range: The range of a relation $R$ is the set of all second elements in the ordered pairs, denoted as $R(D(R))$.

• Range of R: To find the range of $R$, we need to identify the second element in each ordered pair.

  • $(-9,20)$: The second element is $20$.
  • $(-3,10)$: The second element is $10$.
  • $(0,13)$: The second element is $13$.
  • $(2,-10)$: The second element is $-10$.
  • $(8,-3)$: The second element is $-3$.

• Range of R: The range of $R$ is the set of all these second elements, which is $\{20,10,13,-10,-3\}$.

Is R a Function?

A relation $R$ is a function if every first element in the ordered pairs has a unique second element. In other words, a relation $R$ is a function if for every $x$ in the domain, there is a unique $y$ in the range such that $(x,y)$ is in $R$.

• Definition of Function: A relation $R$ is a function if for every $x$ in the domain, there is a unique $y$ in the range such that $(x,y)$ is in $R$.

• Is R a Function?: To determine if $R$ is a function, we need to check if every first element in the ordered pairs has a unique second element.

  • $-9$ has a unique second element, which is $20$.
  • $-3$ has a unique second element, which is $10$.
  • $0$ has a unique second element, which is $13$.
  • $2$ has a unique second element, which is $-10$.
  • $8$ has a unique second element, which is $-3$.

• Is R a Function?: Since every first element in the ordered pairs has a unique second element, $R$ is a function.

Is R One-to-One?

A relation $R$ is one-to-one if every first element in the ordered pairs has a unique second element and every second element in the ordered pairs has a unique first element. In other words, a relation $R$ is one-to-one if for every $x$ in the domain, there is a unique $y$ in the range such that $(x,y)$ is in $R$ and for every $y$ in the range, there is a unique $x$ in the domain such that $(x,y)$ is in $R$.

• Definition of One-to-One: A relation $R$ is one-to-one if for every $x$ in the domain, there is a unique $y$ in the range such that $(x,y)$ is in $R$ and for every $y$ in the range, there is a unique $x$ in the domain such that $(x,y)$ is in $R$.

• Is R One-to-One?: To determine if $R$ is one-to-one, we need to check if every first element in the ordered pairs has a unique second element and every second element in the ordered pairs has a unique first element.

  • $-9$ has a unique second element, which is $20$.

  • $-3$ has a unique second element, which is $10$.

  • $0$ has a unique second element, which is $13$.

  • $2$ has a unique second element, which is $-10$.

  • $8$ has a unique second element, which is $-3$.

  • $20$ has a unique first element, which is $-9$.

  • $10$ has a unique first element, which is $-3$.

  • $13$ has a unique first element, which is $0$.

  • $-10$ has a unique first element, which is $2$.

  • $-3$ has a unique first element, which is $8$.

• Is R One-to-One?: Since every first element in the ordered pairs has a unique second element and every second element in the ordered pairs has a unique first element, $R$ is one-to-one.

Is R Onto?

A relation $R$ is onto if for every $y$ in the range, there is an $x$ in the domain such that $(x,y)$ is in $R$. In other words, a relation $R$ is onto if for every $y$ in the range, there is an $x$ in the domain such that $(x,y)$ is in $R$.

• Definition of Onto: A relation $R$ is onto if for every $y$ in the range, there is an $x$ in the domain such that $(x,y)$ is in $R$.

• Is R Onto?: To determine if $R$ is onto, we need to check if for every $y$ in the range, there is an $x$ in the domain such that $(x,y)$ is in $R$.

  • $20$ is in the range, and $-9$ is in the domain such that $(-9,20)$ is in $R$.
  • $10$ is in the range, and $-3$ is in the domain such that $(-3,10)$ is in $R$.
  • $13$ is in the range, and $0$ is in the domain such that $(0,13)$ is in $R$.
  • $-10$ is in the range, and $2$ is in the domain such that $(2,-10)$ is in $R$.
  • $-3$ is in the range, and $8$ is in the domain such that $(8,-3)$ is in $R$.

• Is R Onto?: Since for every $y$ in the range, there is an $x$ in the domain such that $(x,y)$ is in $R$, $R$ is onto.

Conclusion

Q: What is the relation R?

A: The relation R is a set of ordered pairs that define a connection between two sets. In this case, the relation R is defined by the ordered pairs:

$R=\{(-9,20),(-3,10),(0,13),(2,-10),(8,-3)\}$

Q: What is the domain of R?

A: The domain of R 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.

• Domain of R: The domain of R is $\{-9,-3,0,2,8\}$.

Q: What is the range of R?

A: The range of R 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.

• Range of R: The range of R is $\{20,10,13,-10,-3\}$.

Q: Is R a function?

A: A relation R is a function if every first element in the ordered pairs has a unique second element. In other words, a relation R is a function if for every x in the domain, there is a unique y in the range such that (x,y) is in R.

• Is R a function?: Yes, R is a function because every first element in the ordered pairs has a unique second element.

Q: Is R one-to-one?

A: A relation R is one-to-one if every first element in the ordered pairs has a unique second element and every second element in the ordered pairs has a unique first element. In other words, a relation R is one-to-one if for every x in the domain, there is a unique y in the range such that (x,y) is in R and for every y in the range, there is a unique x in the domain such that (x,y) is in R.

• Is R one-to-one?: Yes, R is one-to-one because every first element in the ordered pairs has a unique second element and every second element in the ordered pairs has a unique first element.

Q: Is R onto?

A: A relation R is onto if for every y in the range, there is an x in the domain such that (x,y) is in R. In other words, a relation R is onto if for every y in the range, there is an x in the domain such that (x,y) is in R.

• Is R onto?: Yes, R is onto because for every y in the range, there is an x in the domain such that (x,y) is in R.

Q: What are the properties of R?

A: The properties of R are:

• Domain of R: The domain of R is $\{-9,-3,0,2,8\}$.
• Range of R: The range of R is $\{20,10,13,-10,-3\}$.
• Is R a function?: Yes, R is a function.
• Is R one-to-one?: Yes, R is one-to-one.
• Is R onto?: Yes, R is onto.

Q: What is the significance of R?

A: The relation R is significant because it demonstrates the properties of a function, one-to-one, and onto relations. It also shows how to determine the domain, range, and other properties of a relation.

Q: How can R be used in real-world applications?

A: The relation R can be used in real-world applications such as:

• Mathematics: R can be used to demonstrate the properties of functions, one-to-one, and onto relations.
• Computer Science: R can be used to represent the relationships between variables in a program.
• Data Analysis: R can be used to analyze data and identify patterns and relationships.

Q: What are the limitations of R?

A: The limitations of R are:

• Limited scope: R is a simple relation and may not be applicable to more complex situations.
• Limited domain: R is defined for a specific domain and may not be applicable to other domains.
• Limited range: R is defined for a specific range and may not be applicable to other ranges.