The Relation R R R Is Defined By The Ordered Pairs Listed Below.$R = {(-3,5), (-3,-1), (-5,15), (10,-6), (10,14)}$1. The Domain Of R R R Is $\square$2. The Range Of R R R Is $\square$3. Is R R R

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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: $R = \{(-3,5), (-3,-1), (-5,15), (10,-6), (10,14)\}$. In this article, we will explore the domain and range of the relation $R$, and discuss whether it is reflexive, symmetric, transitive, and an equivalence relation.

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.

• $(-3,5)$: The first element is $-3$.
• $(-3,-1)$: The first element is $-3$.
• $(-5,15)$: The first element is $-5$.
• $(10,-6)$: The first element is $10$.
• $(10,14)$: The first element is $10$.

The domain of $R$ is the set of all these first elements: $\{-3, -5, 10\}$.

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.

• $(-3,5)$: The second element is $5$.
• $(-3,-1)$: The second element is $-1$.
• $(-5,15)$: The second element is $15$.
• $(10,-6)$: The second element is $-6$.
• $(10,14)$: The second element is $14$.

The range of $R$ is the set of all these second elements: $\{5, -1, 15, -6, 14\}$.

Is $R$ Reflexive?

A relation $R$ is reflexive if every element in the domain is related to itself. In other words, for every $a$ in the domain, there exists an ordered pair $(a, a)$ in $R$. Let's check if $R$ is reflexive.

• The domain of $R$ is $\{-3, -5, 10\}$.
• We need to check if each element in the domain is related to itself.
• There is no ordered pair $(-3, -3)$ in $R$.
• There is no ordered pair $(-5, -5)$ in $R$.
• There is no ordered pair $(10, 10)$ in $R$.

Since there is no ordered pair $(a, a)$ for every $a$ in the domain, $R$ is not reflexive.

Is $R$ Symmetric?

A relation $R$ is symmetric if for every ordered pair $(a, b)$ in $R$, there exists an ordered pair $(b, a)$ in $R$. Let's check if $R$ is symmetric.

• The ordered pair $(-3, 5)$ is in $R$.
• There is no ordered pair $(5, -3)$ in $R$.
• The ordered pair $(-3, -1)$ is in $R$.
• There is no ordered pair $(-1, -3)$ in $R$.
• The ordered pair $(-5, 15)$ is in $R$.
• There is no ordered pair $(15, -5)$ in $R$.
• The ordered pair $(10, -6)$ is in $R$.
• There is no ordered pair $(-6, 10)$ in $R$.
• The ordered pair $(10, 14)$ is in $R$.
• There is no ordered pair $(14, 10)$ in $R$.

Since there is no ordered pair $(b, a)$ for every ordered pair $(a, b)$ in $R$, $R$ is not symmetric.

Is $R$ Transitive?

A relation $R$ is transitive if for every ordered pairs $(a, b)$ and $(b, c)$ in $R$, there exists an ordered pair $(a, c)$ in $R$. Let's check if $R$ is transitive.

• There is no ordered pair $(a, b)$ and $(b, c)$ in $R$ such that $a \neq b$ and $b \neq c$.

Since there is no ordered pair $(a, c)$ for every ordered pairs $(a, b)$ and $(b, c)$ in $R$, $R$ is not transitive.

Is $R$ an Equivalence Relation?

An equivalence relation is a relation that is reflexive, symmetric, and transitive. Since $R$ is not reflexive, symmetric, and transitive, it is not an equivalence relation.

Conclusion

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Q: What is the domain of the relation $R$?

A: The domain of the relation $R$ is the set of all first elements in the ordered pairs. In this case, the domain of $R$ is $\{-3, -5, 10\}$.

Q: What is the range of the relation $R$?

A: The range of the relation $R$ is the set of all second elements in the ordered pairs. In this case, the range of $R$ is $\{5, -1, 15, -6, 14\}$.

Q: Is the relation $R$ reflexive?

A: No, the relation $R$ is not reflexive. A relation is reflexive if every element in the domain is related to itself. In this case, there is no ordered pair $(a, a)$ for every $a$ in the domain.

Q: Is the relation $R$ symmetric?

A: No, the relation $R$ is not symmetric. A relation is symmetric if for every ordered pair $(a, b)$ in $R$, there exists an ordered pair $(b, a)$ in $R$. In this case, there is no ordered pair $(b, a)$ for every ordered pair $(a, b)$ in $R$.

Q: Is the relation $R$ transitive?

A: No, the relation $R$ is not transitive. A relation is transitive if for every ordered pairs $(a, b)$ and $(b, c)$ in $R$, there exists an ordered pair $(a, c)$ in $R$. In this case, there is no ordered pair $(a, c)$ for every ordered pairs $(a, b)$ and $(b, c)$ in $R$.

Q: Is the relation $R$ an equivalence relation?

A: No, the relation $R$ is not an equivalence relation. An equivalence relation is a relation that is reflexive, symmetric, and transitive. Since $R$ is not reflexive, symmetric, and transitive, it is not an equivalence relation.

Q: What are some common mistakes when working with relations?

A: Some common mistakes when working with relations include:

• Assuming that a relation is reflexive, symmetric, or transitive without checking.
• Not checking for the existence of ordered pairs in the relation.
• Not considering the domain and range of the relation.

Q: How can I practice working with relations?

A: You can practice working with relations by:

• Creating your own relations and checking for reflexivity, symmetry, and transitivity.
• Working with different types of relations, such as equivalence relations and partial orders.
• Using online resources and practice problems to help you understand and work with relations.

Q: What are some real-world applications of relations?

A: Relations have many real-world applications, including:

• Database management: Relations are used to define the relationships between data in a database.
• Social network analysis: Relations are used to study the connections between individuals in a social network.
• Computer science: Relations are used to define the relationships between objects in a program.

Q: How can I use relations in my own work or studies?

A: You can use relations in your own work or studies by:

• Applying the concepts of reflexivity, symmetry, and transitivity to real-world problems.
• Using relations to model and analyze complex systems.
• Working with different types of relations to solve problems and answer questions.