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Chapter 9: Problem 36

Give a poset that has a) a minimal element but no maximal element. b) a maximal element but no minimal element. c) neither a maximal nor a minimal element.

Short Answer

Expert verified
a) Natural numbers, b) Negative integers, c) Integers.

Step by step solution

01

Understanding Minimal and Maximal Elements

A minimal element in a poset is an element that is not greater than any other element in the set. A maximal element is one that is not less than any other element in the set.
02

Poset with Minimal Element but No Maximal Element

Consider the set of natural numbers \(\textbf{N} \) with the usual ordering. In this poset, the element '1' is a minimal element because there is no element smaller than 1. However, there is no maximal element because you can always find a larger natural number.
03

Poset with Maximal Element but No Minimal Element

Consider the set of negative integers \(-\textbf{Z}^-\) with the usual ordering. In this poset, the element '-1' is a maximal element because there is no element larger (more positive) than '-1'. There is no minimal element because you can always find a smaller (more negative) integer.
04

Poset with Neither Minimal Nor Maximal Element

Consider the set of all integers \(\textbf{Z} \) with the usual ordering. In this poset, there is neither a minimal element nor a maximal element because for any given integer, you can always find a smaller and a larger one.

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Key Concepts

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

Minimal Element
A minimal element in a partially ordered set (poset) is an element that has no other elements less than it. Imagine a simple scenario where you have a set of natural numbers arranged in their usual order: 1, 2, 3, and so on. Here, the number 1 is a minimal element because there's no natural number smaller than 1. To identify a minimal element, look for an element in your set that doesn't have any predecessors.
Minimal elements are crucial in certain types of mathematical structures and problems. For example, they help in finding the least element or starting point in optimization problems or search algorithms.
  • Minimal elements are unique in well-defined sets but not in general posets.
  • There can be more than one minimal element in a poset.
  • Minimal elements should not be confused with the least element, which is smaller than all other elements in the set.
Maximal Element
A maximal element is the opposite of a minimal element. This is an element in a poset that has no other elements greater than it. For example, consider the set of negative integers ordered naturally: -1, -2, -3, etc. Here, '-1' is a maximal element because there's no negative integer greater than '-1'.
Finding maximal elements is essential in various fields such as graph theory, where identifying the highest node can influence the traversal of graphs.
  • Maximal elements can also be unique but do not have to be.
  • Multiple maximal elements can exist in a given poset.
  • A maximal element should not be confused with a greatest element, which is greater than all other elements in the set.
Partially Ordered Set (Poset)
A partially ordered set, or poset, is a set combined with a partial order. This means that not every pair of elements must be comparable.
Posets are generalizations of the more familiar concept of a totally ordered set (or linearly ordered set), where every pair of elements can be compared.
To better understand a poset, consider the set of natural numbers with the standard ordering: 1, 2, 3, etc.
  • Each natural number can be compared with another to see which is larger or smaller, fulfilling the partial order condition.
However, if we consider another set where the elements are people and the order is based on their height, not all pairs of people would be directly comparable, because some people may have the same height.
  • Posets are represented using Hasse diagrams to visualize the order relations.
  • In Hasse diagrams, minimal and maximal elements can be easily identified.
  • Posets allow for multiple minimal and maximal elements, unlike totally ordered sets where there鈥檚 only one minimum and one maximum.
Understanding posets is fundamental in many branches of mathematics including algebra and data science.

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Most popular questions from this chapter

Let \(R\) be the relation on the set of all people who have visited a particular Web page such that \(x R y\) if and only if person \(x\) and person \(y\) have followed the same set of links starting at this Web page (going from Web page to Web page until they stop using the Web). Show that \(R\) is an equivalence relation.

Which of these collections of subsets are partitions of the set of bit strings of length 8? a) the set of bit strings that begin with 1, the set of bit strings that begin with 00, and the set of bit strings that begin with 01 b) the set of bit strings that contain the string 00, the set of bit strings that contain the string 01, the set of bit strings that contain the string 10, and the set of bit strings that contain the string 11 c) the set of bit strings that end with 00, the set of bit strings that end with 01, the set of bit strings that end with 10, and the set of bit strings that end with 11 d) the set of bit strings that end with 111, the set of bit strings that end with 011, and the set of bit strings that end with 00 e) the set of bit strings that contain 3k ones for some nonnegative integer k, the set of bit strings that contain 3k + 1 ones for some nonnegative integer k, and the set of bit strings that contain 3k + 2 ones for some nonnegative integer k.

Let \(R\) be the relation on the set \(\\{1,2,3,4,5\\}\) containing the ordered pairs \((1,1),(1,2),(1,3),(2,3),(2,4),(3,1),\) \((3,4),(3,5),(4,2),(4,5),(5,1),(5,2),\) and \((5,4) .\) Find \(\begin{array}{llll}{\text { a) } R^{2}} & {\text { b) } R^{3} .} & {\text { c) } R^{4}} & {\text { d) } R^{5}}\end{array}\)

Find the lexicographic ordering of these \(n\) -tuples: a) \((1,1,2),(1,2,1)\) b) \((0,1,2,3),(0,1,3,2)\) c) \((1,0,1,0,1),(0,1,1,1,0)\)

How many different relations are there from a set with \(m\) elements to a set with \(n\) elements?
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