/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 356 If we have an equivalence relati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 356

If we have an equivalence relation that divides a set with \(k\) elements up into equivalence classes each of size \(m,\) what is the number \(n\) of equivalence classes? Explain why.

Short Answer

Expert verified
\( n = \frac{k}{m} \)

Step by step solution

01

Understanding the Problem

An equivalence relation divides a set into equivalence classes. We need to determine the number of these classes given the total number of elements in the set and the size of each class.
02

Define Variables

Let the total number of elements in the set be represented by the variable \(k\) and the size of each equivalence class by the variable \(m\).
03

Equation Setup

The number of equivalence classes, denoted by \(n\), is how many groups of size \(m\) can be formed from \(k\) elements. This relationship can be defined by the equation: \( n = \frac{k}{m} \).
04

Solving the Equation

To find \(n\), solve the equation: \( n = \frac{k}{m} \). Here, \(n\) must be an integer because you can't have a fraction of an equivalence class.
05

Explanation

The division \( \frac{k}{m} \) must be exact, meaning \(k\) should be divisible by \(m\) without leaving a remainder, ensuring that each element is part of one and only one equivalence class.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Equivalence Classes
An equivalence class is a subset formed by an equivalence relation. Imagine a set of students. If they are grouped based on having the same favorite color, each group of students sharing the same favorite color is an equivalence class.

Equivalence classes are crucial because they partition a set into distinct, non-overlapping subsets where each element of the original set belongs to one and only one subset.
For instance, if you have a set of numbers, and your equivalence relation is 'having the same remainder when divided by 3,' the set will be divided into equivalence classes like {0, 3, 6,...}, {1, 4, 7,...}, and {2, 5, 8,...}.

Understanding equivalence relations and classes helps in organizing data into meaningful categories that make analyzing and drawing conclusions easier.
Set Theory
Set theory is the mathematical study of collections of objects, called sets. These objects can be anything: numbers, letters, symbols, or even other sets.

A set is usually denoted by curly braces with the elements inside. For example, a set of the first three natural numbers is written as \{1, 2, 3\}.

With set theory, we use various operations to combine, compare, and arrange sets. Here are some key concepts:
  • Union: Combines all elements from multiple sets. Denoted by \( A \cup B \).
  • Intersection: Finds common elements between sets. Denoted by \( A \cap B \).
  • Difference: Elements in one set but not in another. Denoted by \( A - B \).
Set theory forms the foundation of various mathematical areas and is essential for understanding functions, sequences, and probability.
Division in Mathematics
Division is one of the four fundamental arithmetic operations. In simple terms, when you divide a number, you split it into equal parts.

For example, dividing 12 by 4 means dividing 12 into 4 equal parts, which results in 3. Mathematically, this is expressed as \720:=4.

When it comes to sets and equivalence classes, division helps determine how many groups (equivalence classes) of a specific size (number of elements) can be formed from a larger set.

If you have a set with \k elements and each equivalence class needs to have \m elements, you divide \k by \m, providing you the number of equivalence classes, \. Mathematically, \( n = \frac{k}{m}\).

This division only works perfectly if \k is evenly divisible by \m. Otherwise, the elements cannot be distributed evenly across the equivalence classes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Another relation that you may have learned about in school, perhaps in the guise of "clock arithmetic," is the relation of equivalence modulo \(n\). For integers (positive, negative, or zero) \(a\) and \(b\), we write \(a \equiv b\) \((\bmod n)\) to mean that \(a-b\) is an integer multiple of \(n,\) and in this case, we say that \(a\) is congruent to \(b\) modulo \(n\) and write \(a \equiv b(\bmod n) . .\) Show that the relation of congruence modulo \(n\) is an equivalence relation.

Draw the digraph of the relation from the set \(\\{\mathrm{A}, \mathrm{M}, \mathrm{P}, \mathrm{S}\\}\) to the set \\{Sam, Mary, Pat, Ann, Polly, Sarah\\} given by "is the first letter of."

Now suppose again we are choosing three distinct flavors of ice cream out of four, but instead of putting scoops in a cone or choosing pints, we are going to have the three scoops arranged symmetrically in a circular dish. Similarly to choosing three pints, we can describe a selection of ice cream in terms of which one goes in the dish first, which one goes in second (say to the right of the first), and which one goes in third (say to the right of the second scoop, which makes it to the left of the first scoop). But again, two of these lists will sometimes be equivalent. Once they are in the dish, we can't tell which one went in first. However, there is a subtle difference between putting each flavor in its own small dish and putting all three flavors in a circle in a larger dish. Think about what makes the lists of flavors equivalent, and draw the directed graph whose vertices consist of all lists of three of the flavors of ice cream and whose edges connect two lists that we cannot tell the difference between as dishes of ice cream. How many dishes of ice cream can we distinguish from one another? (h)

Two simple graphs on the set \([n]=\\{1,2, \ldots, n\\}\) with edge sets \(E\) and \(E^{\prime}\) (which we think of a sets of two-element sets for this problem) are said to be isomorphic if there is a permutation \(\sigma\) of \([n]\) which, in its action of two-element sets, carries \(E\) to \(E^{\prime} .\) We say two graphs are different if they are not isomorphic. Thus the number of different graphs is the number of orbits of the set of all two-element subsets of \([n]\) under the action of the group \(S_{n} .\) We can represent an edge set by its characteristic function (as in problem 33). That is we define $$ \chi_{E}(\\{u, v\\})=\left\\{\begin{array}{ll} 1 & \text { if }\\{u, v\\} \in E \\ 0 & \text { otherwise } \end{array}\right. $$ Thus we can think of the set of graphs as a set of colorings with colors 0 and 1 of the set of all two-element subsets of \([n] .\) The number of different graphs with vertex set \([n]\) is thus the number of orbits of this set of colorings under the action of the symmetric group \(S_{n}\) on the set of two- element subsets of \([n] .\) Use this to find the number of different graphs on five vertices.

Explain why the set of all permutations of four elements is a permutation group. How many elements does this group have? This group is called the symmetric group on four letters and is denoted by \(S_{4}\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.