Determining Whether a Set is a Relation or a Function

Oct 13, 2025 by miodeya.com 112 views

In mathematics, a relation is a set of ordered pairs that describe the relationship between two sets. On the other hand, a function is a special type of relation where each input is associated with exactly one output. In this article, we will explore the concept of relations and functions, and determine whether the given set is a relation, a function, or neither.

What is a Relation?

A relation is a set of ordered pairs that describe the relationship between two sets. It is a way of describing how the elements of one set are related to the elements of another set. Relations can be represented using a table, where the rows represent the input values and the columns represent the output values.

Example of a Relation

Consider the following relation:

Input	Output
1	2
2	4
3	6

In this example, the relation is a set of ordered pairs that describe the relationship between the input values and the output values. Each input value is associated with exactly one output value.

What is a Function?

A function is a special type of relation where each input is associated with exactly one output. In other words, a function is a relation where each input value is mapped to a unique output value. Functions can be represented using a table, where the rows represent the input values and the columns represent the output values.

Example of a Function

Consider the following function:

Input	Output
1	2
2	4
3	6

In this example, the function is a special type of relation where each input value is associated with exactly one output value. Each input value is mapped to a unique output value.

Determining Whether a Set is a Relation or a Function

To determine whether a set is a relation or a function, we need to check if each input value is associated with exactly one output value. If each input value is associated with exactly one output value, then the set is a function. If each input value is associated with more than one output value, then the set is not a function.

Given Set

The given set is:

Input	Output
-1	0
0	1
-1	2
-8	4
9	8

Is the Given Set a Relation or a Function?

To determine whether the given set is a relation or a function, we need to check if each input value is associated with exactly one output value. Let's examine the given set:

The input value -1 is associated with two output values: 0 and 2.
The input value 0 is associated with one output value: 1.
The input value -8 is associated with one output value: 4.
The input value 9 is associated with one output value: 8.

Since the input value -1 is associated with two output values, the given set is not a function. Therefore, the correct answer is B. Neither.

Conclusion

In conclusion, a relation is a set of ordered pairs that describe the relationship between two sets, while a function is a special type of relation where each input is associated with exactly one output. To determine whether a set is a relation or a function, we need to check if each input value is associated with exactly one output value. If each input value is associated with exactly one output value, then the set is a function. If each input value is associated with more than one output value, then the set is not a function. In this article, we determined that the given set is not a function, and therefore the correct answer is B. Neither.

References

[1] "Relations and Functions" by Khan Academy
[2] "Relations and Functions" by Math Open Reference
[3] "Relations and Functions" by Wolfram MathWorld

Glossary

Relation: A set of ordered pairs that describe the relationship between two sets.
Function: A special type of relation where each input is associated with exactly one output.
Input: The value that is used to determine the output value.
Output: The value that is determined by the input value.
Ordered Pair: A pair of values that are used to describe the relationship between two sets.
Frequently Asked Questions (FAQs) about Relations and Functions ====================================================================

In this article, we will answer some of the most frequently asked questions about relations and functions.

Q: What is the difference between a relation and a function?

A: A relation is a set of ordered pairs that describe the relationship between two sets, while a function is a special type of relation where each input is associated with exactly one output.

Q: How do I determine whether a set is a relation or a function?

A: To determine whether a set is a relation or a function, you need to check if each input value is associated with exactly one output value. If each input value is associated with exactly one output value, then the set is a function. If each input value is associated with more than one output value, then the set is not a function.

Q: What is an example of a relation that is not a function?

A: Consider the following relation:

Input	Output
1	2
2	4
3	6
1	8

In this example, the input value 1 is associated with two output values: 2 and 8. Therefore, this relation is not a function.

Q: What is an example of a function?

A: Consider the following function:

Input	Output
1	2
2	4
3	6

In this example, each input value is associated with exactly one output value. Therefore, this is a function.

Q: Can a function have multiple input values associated with the same output value?

A: No, a function cannot have multiple input values associated with the same output value. By definition, a function is a relation where each input value is associated with exactly one output value.

Q: Can a relation have multiple output values associated with the same input value?

A: Yes, a relation can have multiple output values associated with the same input value. This is one of the key differences between a relation and a function.

Q: What is the domain of a relation or function?

A: The domain of a relation or function is the set of all input values.

Q: What is the range of a relation or function?

A: The range of a relation or function is the set of all output values.

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

A: Yes, the domain and range of a relation or function can be the same. For example, consider the following function:

Input	Output
1	1
2	2
3	3

In this example, the domain and range are the same.

Q: Can a relation or function have an empty domain or range?

A: Yes, a relation or function can have an empty domain or range. For example, consider the following relation:

Input	Output
	1

In this example, the domain is empty.

Conclusion

In conclusion, relations and functions are important concepts in mathematics that describe the relationship between two sets. By understanding the difference between a relation and a function, you can determine whether a set is a relation or a function. We hope that this article has helped to answer some of the most frequently asked questions about relations and functions.

References

[1] "Relations and Functions" by Khan Academy
[2] "Relations and Functions" by Math Open Reference
[3] "Relations and Functions" by Wolfram MathWorld

Glossary

Relation: A set of ordered pairs that describe the relationship between two sets.
Function: A special type of relation where each input is associated with exactly one output.
Input: The value that is used to determine the output value.
Output: The value that is determined by the input value.
Ordered Pair: A pair of values that are used to describe the relationship between two sets.
Domain: The set of all input values.
Range: The set of all output values.