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Chapter 5: Problem 86

Is the set \(\\{(J o h n, A),(M a r y, A)\), (Miguel, A), (Lee, A)\\} a function? Explain.

Short Answer

Expert verified
Yes, it is a function because each input is paired with exactly one output.

Step by step solution

01

Define a Function

A function is a relation between a set of inputs (domain) and a set of possible outputs (codomain) with the property that each input is related to exactly one output.
02

Identify Inputs and Outputs

Identify the inputs and outputs in the set \(\{ (John, A), (Mary, A), (Miguel, A), (Lee, A) \}\). Here, the inputs are John, Mary, Miguel, and Lee. The output is A.
03

Check Uniqueness of Outputs

Verify that each input in the set is related to exactly one output. In the given set, each input (John, Mary, Miguel, Lee) is paired with exactly one output (A).
04

Conclusion

Since every input has exactly one output, the set \(\{ (John, A), (Mary, A), (Miguel, A), (Lee, A) \}\) satisfies the definition of a function.

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Key Concepts

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

Relation
In mathematics, a relation is simply a connection or association between elements of one set and elements of another set. It can be represented as a set of ordered pairs \t(one element from each set). For example, in our case: \(\{(John, A), (Mary, A), (Miguel, A), (Lee, A)\}\), the relation shows how each person (John, Mary, Miguel, Lee) is associated with 'A'. Not all relations are functions, but all functions are a special kind of relation.
Domain
The domain of a function is the set of all possible inputs. In simpler terms, it's the set of all first elements in our ordered pairs. For the set \(\{(John, A), (Mary, A), (Miguel, A), (Lee, A)\}\), the domain consists of \(\{John, Mary, Miguel, Lee\}\). Every individual in this set is an input to the function.
Codomain
The codomain of a function is the set of all possible outputs that could potentially be paired with the inputs. Just like the domain is the set of first elements, the codomain includes the second elements of our pairs. In our example, the codomain is relatively simple: \(\{A\}\). It means that 'A' is the only possible output for the inputs in our set.
Inputs
Inputs in a function are the specific values or elements from the domain that we use to get a result. They are the 'independent' part of our relationship. For \(\{(John, A), (Mary, A), (Miguel, A), (Lee, A)\}\), the inputs are John, Mary, Miguel, and Lee. Each one of these names is an element we plug into our function.
Outputs
Outputs are the results or values that we get from our inputs. They come from the codomain and are the 'dependent' part of our relationship. Here, each input (John, Mary, Miguel, Lee) gives us the same output, which is 'A'. This is why our set is a function. Each input corresponds to exactly one output, following the rule that defines a function.

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