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Chapter 3: Problem 17

Identify the set as a relation, a function, or both a relation and a function. \(\\{(\mathrm{MN}\), state \()\), (VA, state), (LWS, airport),(SEA, airport \()\\}\)

Short Answer

Expert verified
The set is both a relation and a function.

Step by step solution

01

Understand Definitions

A relation is any set of ordered pairs. A function is a specific type of relation where each input (first element) is related to exactly one output (second element).
02

Identify Ordered Pairs

List the ordered pairs given in the set: (MN, state), (VA, state), (LWS, airport), (SEA, airport).
03

Check for Relation

Since the given set contains ordered pairs, it is a relation by definition.
04

Check for Function

To determine if it's a function, check if each input (first element) appears only once: MN appears once, VA appears once, LWS appears once, SEA appears once. Since there are no repeating first elements, each input maps to one and only one output.
05

Conclusion

Since the set meets the criteria of both a relation and a function, it is both a relation and a function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
In algebra, an ordered pair is a fundamental concept. An ordered pair is a pair of elements written in a specific order, usually as \((a, b)\). The first element is called the 'input' and the second element is the 'output'. This order matters because \((a, b)\) is different from \((b, a)\).
For example, in the set \((MN, state), (VA, state), (LWS, airport), (SEA, airport)\), each pair establishes a connection or mapping between two elements.
Recognizing ordered pairs and their significance is crucial when dealing with relations and functions in algebra.
Function Definition
A function is a special type of relation. In a function, every input must map to exactly one output. This means that if you have an input, there's only one corresponding output. No input can have two different outputs.
To determine if a relation is a function, check each ordered pair. Ensure that each first element (input) appears only once.
In our example set, \(MN, state), (VA, state), (LWS, airport), (SEA, airport)\), each input (MN, VA, LWS, SEA) maps to only one output. Hence, this set is both a function and a relation.
Relation Definition
A relation in algebra is any set of ordered pairs. It鈥檚 a broader concept compared to a function. All functions are relations, but not all relations are functions.
In a relation, an input can map to multiple outputs. But for a function, each input must map to one and only one output.
In our given set \(MN, state), (VA, state), (LWS, airport), (SEA, airport)\), the presence of ordered pairs means it qualifies as a relation. Regardless of whether the inputs map to one or multiple outputs, if the set includes ordered pairs, it is a relation.
Input-Output Mapping
Input-output mapping describes how each input from a set relates to an output from another set. Understanding this mapping is key to distinguishing between relations and functions.
In the mapping process, every input must correspond to an output. In functions, each input has a unique output, ensuring a one-to-one or many-to-one relationship.
For example, in the set \(MN, state), (VA, state), (LWS, airport), (SEA, airport)\), the inputs (MN, VA, LWS, SEA) map to their respective outputs (state, state, airport, airport). This shows a clear and singular mapping for each input, validating both relation and function criteria.

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Most popular questions from this chapter

Some learning preferences describe how you prefer to receive, think about, and learn new information. These preferences include visual learning, auditory learning, and kinesthetic learning. Many students use more than one of these categories as they learn mathematics. \- Visual learners prefer to see information. Although you definitely listen to your instructor, you also like to see the example on a white board or screen. You may be able to recall a process by visualizing it in your mind; you may learn better by organizing information in charts, tables, diagrams, or pictures. You may prefer the use of colored markers instead of just black. \- Auditory learners prefer to hear information. Although you definitely watch what your instructor is doing, you also like your instructor to explain things aloud as he or she works. You may find it difficult to take notes because you cannot concentrate enough on what is being said while you write. You may learn better if you have the chance to explain things to others. \- Kinesthetic learners prefer to do. You may find it difficult to sit still and just watch and listen; you want to be trying it out. You may find that you must take notes in order to learn. If you only watch and listen, you may understand the concept but not remember it after you leave the classroom. You often learn better if you can show others how to do things. Have you noticed anything that your instructor does while teaching that you find helps you remember what has been taught?

For exercises 9鈥�20, (a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,4) ;(3,10)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(-8, \frac{1}{4}\right) ;\left(16, \frac{1}{2}\right)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(0, \frac{3}{4}\right) ;\left(\frac{1}{4}, 0\right)\)

Use the slope formula to find the slope of the line that passes through the points. \((0,-9) ;(-2,0)\)
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