Suppose That The Relation \[$ S \$\] Is Defined As Follows:$\[ S = \{(5, A), (5, C), (1, B)\} \\]Give The Domain And Range Of \[$ S \$\]. Write Your Answers Using Set Notation.Domain = \[$\square\$\]Range =

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What is a Relation?

A relation is a set of ordered pairs that connect elements from one set to another. It is a fundamental concept in mathematics, particularly in algebra and set theory. In this article, we will explore the concept of relations and how to find their domains and ranges.

The Given Relation

The relation { S $}$ is defined as follows:

S={(5,a),(5,c),(1,b)}{ S = \{(5, a), (5, c), (1, b)\} }

This relation consists of three ordered pairs: (5, a), (5, c), and (1, b). Each ordered pair represents a connection between an element from the first set (the domain) and an element from the second set (the range).

Domain of the Relation

The domain of a relation is the set of all elements from the first set that are connected to elements in the second set. In other words, it is the set of all first elements in the ordered pairs.

To find the domain of { S $}$, we need to identify the first elements in each ordered pair:

  • (5, a) - The first element is 5.
  • (5, c) - The first element is 5.
  • (1, b) - The first element is 1.

Since the first elements are 5 and 1, the domain of { S $}$ is:

Domain={5,1}{ \text{Domain} = \{5, 1\} }

Range of the Relation

The range of a relation is the set of all elements from the second set that are connected to elements in the first set. In other words, it is the set of all second elements in the ordered pairs.

To find the range of { S $}$, we need to identify the second elements in each ordered pair:

  • (5, a) - The second element is a.
  • (5, c) - The second element is c.
  • (1, b) - The second element is b.

Since the second elements are a, c, and b, the range of { S $}$ is:

Range={a,c,b}{ \text{Range} = \{a, c, b\} }

Conclusion

In this article, we have explored the concept of relations and how to find their domains and ranges. We have defined the relation { S $}$ and identified its domain and range using set notation. The domain of { S $}$ is {5, 1}, and the range is {a, c, b}. Understanding relations and their domains and ranges is essential in mathematics, particularly in algebra and set theory.

Further Reading

If you want to learn more about relations and their domains and ranges, I recommend checking out the following resources:

Q: What is a relation?

A: A relation is a set of ordered pairs that connect elements from one set to another. It is a fundamental concept in mathematics, particularly in algebra and set theory.

Q: How do I find the domain of a relation?

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

Q: How do I find the range of a relation?

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

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 is the set of all second elements in the ordered pairs.

Q: Can a relation have multiple domains or ranges?

A: No, a relation can only have one domain and one range. However, a relation can have multiple ordered pairs, and each ordered pair can have a different first element (domain) and a different second element (range).

Q: How do I determine if two relations are equal?

A: Two relations are equal if and only if they have the same ordered pairs. In other words, if two relations have the same first and second elements in the same order, then they are equal.

Q: Can a relation be empty?

A: Yes, a relation can be empty. An empty relation is a relation that has no ordered pairs.

Q: What is the importance of relations in mathematics?

A: Relations are important in mathematics because they help us understand how sets are connected. Relations are used in many areas of mathematics, including algebra, set theory, and graph theory.

Q: Can you give an example of a relation?

A: Yes, here is an example of a relation:

R={(2,3),(4,5),(6,7)}{ R = \{(2, 3), (4, 5), (6, 7)\} }

In this example, the relation R has three ordered pairs: (2, 3), (4, 5), and (6, 7). The domain of R is {2, 4, 6}, and the range of R is {3, 5, 7}.

Q: Can you give an example of a relation with multiple domains and ranges?

A: No, a relation cannot have multiple domains and ranges. However, a relation can have multiple ordered pairs, and each ordered pair can have a different first element (domain) and a different second element (range).

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

A: Yes, here is an example of a relation with an empty domain:

R={(1,2),(3,4),(5,6)}{ R = \{(1, 2), (3, 4), (5, 6)\} }

In this example, the relation R has three ordered pairs: (1, 2), (3, 4), and (5, 6). The domain of R is {1, 3, 5}, and the range of R is {2, 4, 6}. However, if we remove the ordered pair (1, 2), then the domain of R becomes empty.

Conclusion

In this article, we have answered some common questions about relations and their domains and ranges. We have defined what a relation is, how to find the domain and range of a relation, and how to determine if two relations are equal. We have also given examples of relations with multiple domains and ranges, and relations with empty domains and ranges.