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Chapter 0: Problem 33

In Exercises 29 through 34, determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation. $$ n \cdot R m \text { in } \mathrm{Z}^{+} \text {if } n \text { and } m \text { have the same number of digits in the usual base ten notation } $$

Short Answer

Expert verified
The relation is an equivalence relation, partitioning positive integers by digit count.

Step by step solution

01

Understand the Relation

The relation \( n \cdot R m \) states that two positive integers \( n \) and \( m \) are related if they have the same number of digits in base ten. This implies that the equivalence classes consist of numbers with the same number of digits.
02

Check Reflexivity

A relation is reflexive if for every element \( n \) in the set, \( n \cdot R n \) holds true. Here, each number has the same number of digits as itself, so the relation is reflexive.
03

Check Symmetry

A relation is symmetric if for any two elements \( n \) and \( m \), if \( n \cdot R m \), then \( m \cdot R n \). Since the number of digits is the same for both directions, the relation is symmetric.
04

Check Transitivity

A relation is transitive if whenever \( n \cdot R m \) and \( m \cdot R p \), then \( n \cdot R p \). If two numbers have the same number of digits as a third number, they must have the same number of digits as each other. Thus, the relation is transitive.
05

Conclusion on Equivalence Relation

Since the relation satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.
06

Describe the Partition

The equivalence relation partitions \( \mathbb{Z}^{+} \) into subsets where each subset contains numbers with the same number of digits: \( \{1,2,3,\ldots,9\} \), \( \{10,11,\ldots,99\} \), \( \{100,101,\ldots,999\} \), and so on.

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Key Concepts

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

Reflexivity
In mathematics, the concept of reflexivity is crucial to understanding equivalence relations. A relation is reflexive if every element within a set is related to itself. Consider the exercise we are exploring: the relation \( n \cdot R m \) between numbers having the same number of digits. Here, reflexivity implies that each number has the identical number of digits as itself. Therefore, the condition is trivially met for all numbers since, clearly, any digit count compared to itself remains unchanged. This property ensures that all numbers individually satisfy this relation, confirming reflexivity.
Symmetry
Symmetry in relations is a vital concept where if element \( n \) is related to element \( m \), then \( m \) should also be related to \( n \). In our example, symmetry is easy to verify. If two numbers, \( n \) and \( m \), have the same number of digits, then it is undeniably true that \( m \) has the same number of digits as \( n \). The direction of comparison is irrelevant because the equality between their digit counts holds in both directions. This bi-directional nature of the relation confirms that symmetry exists in the context of our exercise.
Transitivity
Transitivity is another essential feature of equivalence relations. It requires that if a relation holds between \( n \) and \( m \), and also between \( m \) and \( p \), then it must hold between \( n \) and \( p \). For our digits-based relation, consider numbers \( n \), \( m \), and \( p \) where \( n \) and \( m \) share the same digit count, and \( m \) and \( p \) do as well. Inevitably, \( n \) and \( p \) will also have the same digit count, reinforcing the relation's transitive property. This transitive nature ensures the continuous flow of the relation, bridging all connected elements into a unified equivalence class.
Equivalence Classes
Equivalence classes arise from dividing a set into distinct groups based on an equivalence relation. Each class contains elements that are equivalent to each other. In our exercise, all positive integers are grouped based on having identical digit counts. For instance, the class containing the numbers \( 1 \) through \( 9 \) forms one class, as all have one digit. Similarly, numbers from \( 10 \) to \( 99 \) form another class, all sharing two digits. Each class represents a collective that meets the conditions of reflexivity, symmetry, and transitivity, thus solidifying its role in an equivalence relation.
Partitions in Mathematics
In mathematics, partitions break down a set into non-overlapping subsets where every element is included in exactly one subset. The concept is elegantly illustrated by equivalence relations, as seen in our exercise. The integers \( \mathbb{Z}^+ \) can be divided into subsets such as \( \{1, 2, ..., 9\} \), \( \{10, 11, ..., 99\} \), based on their digit lengths. Each subset represents an equivalence class, and together they form a partition of the entire set. This partitioning ensures that each number belongs to one distinct group, making the analysis of relations both systematic and comprehensive.

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Determine whether the given relation is an equivalence relation on the set. Describe the, partition arising from each equivalence relation. $$ x \cdot \mathcal{R} y \text { in } \mathbb{R} \text { if }|x|=|y| $$
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