/*! 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 52 Give an example of an infinite l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 52

Give an example of an infinite lattice with a) neither a least nor a greatest element. b) a least but not a greatest element. c) a greatest but not a least element. d) both a least and a greatest element.

Short Answer

Expert verified
a) \(\textbf{Z}\), b) \(\textbf{Z}^+\), c) \(\textbf{Z}^-\), d) \[0, 1\].

Step by step solution

01

Understanding lattices

A lattice is a partially ordered set in which any two elements have a unique supremum (least upper bound) and an infimum (greatest lower bound).
02

Example for an infinite lattice with neither a least nor a greatest element

Consider the set of all integers \(\textbf{Z}\) with the usual order. It has no least or greatest element because for any integer, there is always a smaller and a larger integer.
03

Example for an infinite lattice with a least but not a greatest element

Consider the set of all positive integers \(\textbf{Z}^+\) with the usual order. The least element is 1, but there is no largest positive integer.
04

Example for an infinite lattice with a greatest but not a least element

Consider the set of all negative integers \(\textbf{Z}^-\) with the usual order. Here, there is no least element, but \(-1\) is the greatest element.
05

Example for an infinite lattice with both a least and a greatest element

Consider the closed interval \[0, 1\] on the real number line \(\textbf{R}\) with the usual order. The least element is 0 and the greatest element is 1.

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.

Partially Ordered Set
A partially ordered set, or poset, is a set combined with a partial order. A partial order is a binary relation that is reflexive, antisymmetric, and transitive.
For example:
  • Reflexive: Every element is related to itself, so \(a \leq a\).
  • Antisymmetric: If \(a \leq b\) and \(b \leq a\), then \(a = b\).
  • Transitive: If \(a \leq b\) and \(b \leq c\), then \(a \leq c\).
In posets, elements don't need to be comparable. This differentiates them from totally ordered sets, where every pair of elements is comparable.
Understanding this aids in comprehending other concepts, such as lattices.
Least Upper Bound
The least upper bound, or supremum, of a subset is the smallest element in the poset that is greater than or equal to every element in the subset.
For example, in the set of integers \(\mathbb{Z}\):
  • The least upper bound of the set \{1, 2, 3\} is 3 because it is the smallest integer greater than or equal to 1, 2, and 3.
Supremum may not exist in all sets. For instance, the set of positive integers \(\mathbb{Z}^+\) has no least upper bound because it has no upper limit.
Greatest Lower Bound
The greatest lower bound, or infimum, of a subset is the largest element in the poset that is less than or equal to every element in the subset.
For example, in the set of integers \(\mathbb{Z}\):
  • The greatest lower bound of the set \{1, 2, 3\} is 1 because it is the largest integer less than or equal to 1, 2, and 3.
Just like the least upper bound, an infimum may not always exist. For instance, in the real numbers \(\mathbb{R}\), the set of positive integers \(\mathbb{Z}^+\) has an infimum, which is 1.
Positive Integers
Positive integers, denoted \(\mathbb{Z}^+\), are all the whole numbers greater than zero: 1, 2, 3, 4, and so on.
In the context of lattices, \(\mathbb{Z}^+\) is an infinite lattice with a least element but no greatest element.
Key properties:
  • Least element: The smallest positive integer is 1.
  • No greatest element: No matter how large an integer is, there's always a larger one.
  • Example: Consider the set \{1, 2, 3\}, its least element is 1, and it goes on infinitely without a highest number.
This makes \(\mathbb{Z}^+\) a useful example when studying sets with these specific attributes.
Negative Integers
Negative integers, denoted \(\mathbb{Z}^-\), are all the whole numbers less than zero: -1, -2, -3, -4, and so on.
In lattice theory, \(\mathbb{Z}^-\) is an infinite lattice with a greatest element but no least element.
Key properties:
  • Greatest element: The largest negative integer is -1.
  • No least element: No matter how small a negative integer is, there's always a smaller one.
  • Example: Consider the set \{-1, -2, -3\}, its greatest element is -1, and it extends infinitely downward.
This is useful for understanding sets where there is a maximum value, but no minimum.
Real Number Line
The real number line, \(\mathbb{R}\), includes all rational and irrational numbers. It's a continuous set of numbers that extends infinitely in both positive and negative directions.
In terms of lattices, we often look at specific intervals. For example, the interval \[0, 1\]:
  • Least element: 0.
  • Greatest element: 1.
  • Example: The set \(\{x \: 0 \leq x \leq 1 \}\) includes all real numbers between 0 and 1, inclusive.
Understanding these intervals helps in various mathematical analyses, including calculus and analysis.

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

a) Show that there is exactly one greatest element of a poset, if such an element exists. b) Show that there is exactly one least element of a poset, if such an element exists.

Determine whether each of these posets is well-ordered. a) \((S, \leq),\) where \(S=\\{10,11,12, \ldots\\}\) b) \((\mathbf{Q} \cap[0,1], \leq)\) (the set of rational numbers between 0 and 1 inclusive) c) \((S, \leq),\) where \(S\) is the set of positive rational numbers with denominators not exceeding 3 d) \(\left(\mathbf{Z}^{-}, \geq\right),\) where \(\mathbf{Z}^{-}\) is the set of negative integers

Determine whether the relation \(R\) on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where \((a, b) \in R\) if and only if a) \(a\) is taller than \(b\). b) \(a\) and \(b\) were born on the same day. c) \(a\) has the same first name as \(b\) . d) \(a\) and \(b\) have a common grandparent.

Draw the Hasse diagram for the less than or equal to relation on \(\\{0,2,5,10,11,15\\} .\)

Suppose that \(R\) and \(S\) are reflexive relations on a set \(A .\) Prove or disprove each of these statements. a) \(R \cup S\) is reflexive. b) \(R \cap S\) is reflexive. c) \(R \oplus S\) is irreflexive. d) \(R-S\) is irreflexive. e) \(S \circ R\) is reflexive.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.