/*! 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 10 Give an example of a poset with ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 10

Give an example of a poset with four maximal elements but no greatest element.

Short Answer

Expert verified
The poset \(P = \{a, b, c, d, e\}\) with the order \(\leq\) defined by \(e \leq a, e \leq b, e \leq c, e \leq d\) is an example of a poset that has four maximal elements (\(a, b, c, d\)) but no greatest element.

Step by step solution

01

- Understand Basic Terminology

A poset (partially ordered set) is a set equipped with a partial order, which means not every pair of elements have to be comparable. For a poset, we call an element 'maximal' if there is no element greater than it, and 'greatest' if it's greater than every other element. In a poset, there can be several maximal elements that aren't comparable with each other, which means no single element can be identified as the greatest.
02

- Create a poset with example

As an example poset, take the partially ordered set \(P = \{a, b, c, d, e\}\), with the order defined by the binary relation \(\leq\) where \(e \leq a, e \leq b, e \leq c, e \leq d\). In this poset, \(a, b, c, d\) are the four maximal elements, as there are no elements greater than any of them. However, they are not comparable with each other, so there is no greatest element.
03

- Validate the Solution

Our partial order needs to satisfy three properties: reflexivity (every element is less than or equal to itself), antisymmetry (if \(a \leq b\) and \(b \leq a\), then \(a = b\)) and transitivity (if \(a \leq b\) and \(b \leq c\), then \(a \leq c\)). In this case, we have ensured that all these properties are met. Hence, the given poset satisfies the conditions of the exercise.

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!

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

Define the relation \(\mathscr{R}\) on the set \(\mathbf{Z}\) by \(a \mathscr{R} b\) if \(a-b\) is a nonnegative even integer. Verify that \(\Re\) defines a partial order for \(\mathbf{Z}\). Is this partial order a total order?

Let \(A=\\{1,2,3,4,5,6\\} \times(1,2,3,4,5,6\\}\). Define \(\$$ on \)A\( by \)\left(x_{1}, y_{1}\right) \mathscr{( x _ { 2 } , y _ { 2 } )}\(, if \)x_{1} y_{1}=x_{2} y_{2}\( a) Verify that \)\mathscr{R}\( is an equivalence relation on \)A\(. b) Determine the equivalence classes \)[(1,1)],[(2,2)]\(, \)[(3,2)]\(, and \)[(4,3)]$.

Let \(A=\\{1,2,3,4,5,6,7\\} .\) How many symmetric relations on \(A\) contain exactly (a) four ordered pairs? (b) five ordered pairs? (c) seven ordered pairs? (d) eight ordered pairs?

Let \(A=\\{1,2,3,4,5,6,7\\}\), For each of the following values of \(r\), determine an equivalence relation \(\mathscr{H}\) on \(A\) with \(|\mathscr{R}|=\) \(r\), or explain why no such relation exists. (a) \(r=6 ;\) (b) \(r=7 ;\) (c) \(r=8 ;\) (d) \(r=9\); (e) \(r=11 ;\) (f) \(r=22\) (h) \(r=30\); (i) \(r=31\) (f) \(r=22 ;\) (g) \(r=23\);

Prove that if a poset ( \(A, \mathscr{R})\) has a least element, it is unique.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.