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Chapter 5: Problem 23

Give a recursive definition of the set of positive integers that are multiples of \(5 .\)

Short Answer

Expert verified
1. Base case: 5 is in the set. 2. If n is in the set, then n + 5 is in the set.

Step by step solution

01

Base Case

Identify the smallest positive integer that is a multiple of 5. The base case is the initial element in the set. The smallest positive multiple of 5 is 5.
02

Recursive Step

Define how to obtain other elements in the set from the base case using recursion. For a number to be a multiple of 5, add 5 to any multiple of 5 already in the set. If n is an element of the set and it is a multiple of 5, then n + 5 is also a multiple of 5 and should be in the set.
03

Formal Recursive Definition

Combine the base case and the recursive step to form a formal definition. The set of positive integers that are multiples of 5 can be defined recursively as:1. Base case: 5 is in the set.2. Recursive step: If n is in the set, then n + 5 is also in the set.

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Key Concepts

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

Base Case
In a recursive definition, the base case is the simplest, initial element in the set. Here, it serves as the starting point for defining more complex elements.

For the set of positive integers that are multiples of 5, the base case is the smallest positive multiple of 5. This number is, of course, 5 itself. By identifying 5 as the base case, we lay the foundation for generating all other elements in the set.

Think of the base case as the seed from which the rest of the set grows. Without this initial point, you have no starting place in the set. Choosing the correct base case is crucial because it ensures that the recursive process begins correctly.
Recursive Step
The recursive step defines how to obtain new elements in the set based on already known elements.

For the set of positive integers that are multiples of 5, the recursive step tells us how to generate the next multiple of 5. If we know a number, say \(n\), is in the set, then we can generate the next number in the set by adding 5 to \(n\).

So, if \(n\) is a multiple of 5, then \(n + 5\) is also a multiple of 5. This process can be repeated infinitely, creating a sequence like 5, 10, 15, and so on. The key here is understanding that each new element is built directly from a previous element, using a simple, repetitive rule.
Positive Integers
Positive integers are the set of whole numbers greater than zero. In mathematical terms, they are denoted as \(\text{\mathbb{Z}}^{+}\).

When we talk about positive integers that are multiples of 5, we are specifically referring to numbers like 5, 10, 15, and so on. Each of these numbers can be described as \(5k\), where \(k\) is a positive integer.

This ensures that all elements are not only multiples of 5 but also fall within the realm of positive values. By focusing on positive integers, we avoid including zero or negative numbers, which would not fit the definition within this context.

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Most popular questions from this chapter

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Let \(a_{1}, a_{2}, \ldots, a_{n}\) be positive real numbers. The arithmetic mean of these numbers is defined by $$ A=\left(a_{1}+a_{2}+\cdots+a_{n}\right) / n $$ and the geometric mean of these numbers is defined by $$ G=\left(a_{1} a_{2} \cdots a_{n}\right)^{1 / n} . $$ Use mathematical induction to prove that \(A \geq G\) .

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