Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 11: Problem 6

There are five different equivalence relations on the set \(A=\\{a, b, c\\} .\) Describe them all. Diagrams will suffice.

Short Answer

Expert verified
The 5 equivalence relations are: (1) all elements are separately related solely to themselves, (2) a and b are related, and c is related only to itself, (3) a and c are related, and b is related only to itself, (4) b and c are related, and a is related only to itself, (5) all the three elements a, b and c are related to each other.

Step by step solution

01

Equivalence Relation 1

The first equivalence relation is the identity relation where, due to reflexivity, each element is only related to itself. We draw three separate loops, each around a different element.
02

Equivalence Relation 2

The second equivalence relation is when two elements are related and the remaining one is only related to itself. This corresponds to two loops: one around the pair of related elements and the other around the remaining individual element.
03

Equivalence Relation 3

This relation is another variation where two elements are related and the remaining is only related to itself. But the pair of related elements is different from the previous equivalence relation. Again, we draw two loops, each enclosing the related elements.
04

Equivalence Relation 4

In the fourth relation, once again, two elements are related and the remaining is only related to itself. But this time, the pair of related elements is different from the first and second relations. We again draw two loops, each encircling the connected elements.
05

Equivalence Relation 5

The fifth relation is when all three elements are related. We can draw one big loop enclosing all three elements.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Suppose \([a],[b] \in \mathbb{Z}_{n},\) and \([a]=\left[a^{\prime}\right]\) and \([b]=\left[b^{\prime}\right] .\) Alice adds \([a]\) and \([b]\) as \([a]+[b]=\) \([a+b] .\) Bob adds them as \(\left[a^{\prime}\right]+\left[b^{\prime}\right]=\left[a^{\prime}+b^{\prime}\right]\). Show that their answers \([a+b]\) and \(\left[a^{\prime}+b^{\prime}\right]\) are the same.

Let \(A=\\{a, b, c, d\\} .\) Suppose \(R\) is the relation $$ \begin{aligned} R=&\\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a),(a, c),(c, a)\\\ &(a, d),(d, a),(b, c),(c, b),(b, d),(d, b),(c, d),(d, c)\\} \end{aligned} $$ Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.

List all the partitions of the set \(A=\\{a, b\\}\). Compare your answer to the answer to Exercise 5 of Section 11.3 .

Modifying Exercise 8 (above) slightly, define a relation \(\sim\) on \(\mathbb{Z}\) as \(x \sim y\) if and only if \(|x-y| \leq 1 .\) Say whether \(\sim\) is reflexive. Is it symmetric? Transitive?

Suppose \(A \neq \varnothing .\) Since \(\varnothing \subseteq A \times A,\) the set \(R=\varnothing\) is a relation on \(A .\) Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.

Sign-up for free