Question

True or False A function is a relation between two sets \(D\) and \(R\) so that each element \(x\) in the first set \(D\) is related to exactly one element \(y\) in the second set \(R\)

Short answer

True

Step-by-step solution

Understand the Definition of a Function

Begin by recalling the definition of a function. A function from a set D (the domain) to a set R (the range) is a rule that assigns each element in D exactly one element in R. In other words, for every input from the domain, there is a unique output in the range.

Identify Key Characteristics

Identify the key characteristics from the definition: 1) Each element in D is related to an element in R. 2) Each element in D is related to exactly one element in R. This means that no element in the domain maps to more than one element in the range.

Analyze the Given Statement

The given statement is: 'A function is a relation between two sets D and R so that each element x in the first set D is related to exactly one element y in the second set R.'

Compare with Definition

Compare the given statement to the definition described in the previous steps. The statement aligns perfectly with the key characteristics of a function mentioned earlier.

Determine Truth Value

Since the given statement matches the definition of a function, it is accurate to classify the statement as True.

Key concepts

Domain and Range

Understanding the concept of a function begins with knowing what 'domain' and 'range' mean. In mathematics, the domain of a function is the complete set of possible values of the independent variable. In other words, it's what you input into the function.
The range, on the other hand, is the set of all possible output values. Imagine you have a function that assigns each person in your class (domain) to the grade they got on a test (range). Each student can only get one grade for that test, so every input from the domain maps to exactly one value in the range.
When talking about domain and range, it helps to visualize it like a mapping. The domain acts as your set of addresses, and the range acts as your set of destinations.

Relation

A 'relation' in math is just a way to pair up elements from one set with elements of another set. If we think about a function as a special type of relation, it has a specific rule: every element in the domain must pair with exactly one element in the range.
To clarify, let's look at different kinds of relations. For instance, in some relations, one input could be paired with several outputs. But in a function, this doesn't happen. Each input (element in the domain) has only one output (element in the range).
For example, if a relation in a school pairs students (domain) with their homeroom teachers (range), each student can only have one homeroom teacher. This makes it a function. If students were paired with all their subject teachers, it would not be a function because an input (student) would map to multiple outputs (teachers).

Unique Mapping

The term 'unique mapping' is essential in understanding functions. It means that every input from the domain must map to one and only one output in the range. This is a key characteristic of functions.
Unique mapping helps ensure that the relationship is predictable and consistent. If we know the input, we can always determine the corresponding output.
To illustrate, consider a simple function like f(x) = 2x. Here, each input value 'x' from the domain maps to twice its value in the range. If you input 3, the unique output will be 2*3 = 6. No other value from the domain will map to 6 within this function.
Thus, unique mapping guarantees that for every x in the domain, there is a single y in the range, upholding the integrity of a function.