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Chapter 10: Problem 43

Draw a Hasse diagram for a partially ordered set that has two maximal elements and two minimal elements and is such that each element is comparable to exactly two other elements.

Short Answer

Expert verified
The Hasse diagram for a partially ordered set that meets the given conditions can be illustrated as follows: Diagram: D E | | C / \ A B Here, A and B are minimal elements, D and E are maximal elements, and each element is comparable to exactly two other elements.

Step by step solution

01

Understand the terms

A Hasse diagram is a graphical representation of a partially ordered set, or poset, to represent the partial order between its elements. The elements are represented as points, and the relation between the elements is shown by edges. If an element "a" is related to the element "b" and there is no other element between them, then "a" is connected to "b" by an edge. A partially ordered set is a set with a binary relation between its elements, which satisfies the following three conditions: 1. Reflexivity: Every element is related to itself (aRa). 2. Antisymmetry: If aRb and bRa, then a = b (meaning that there is only one way to relate elements in the set). 3. Transitivity: If aRb and bRc, then aRc (if you can go from one element to another, then the path will be continuous). Maximal elements are those elements in the poset that are not related to any other elements that are larger than them, and minimal elements are those elements in the poset that are not related to any other elements that are smaller than them. An element is said to be comparable with another if there is a relation between them.
02

Start with the minimal elements

Since we are asked to create a Hasse diagram with two minimal elements, we would start by placing two points, A and B, representing the minimal elements. These two points should not be connected by an edge since they are minimal. Diagram: A B
03

Adding elements to make them comparable

Now, we need to add elements to the diagram such that each element is comparable to exactly two other elements. We know that the points A and B must be minimal elements and thus need to be connected to only one other point each. We now create another vertex, C, and connect both A and B to it. In this way, both A and B are now comparable to exactly one other element (point C): Diagram: C / \ A B
04

Add maximal elements

Since we need to have two maximal elements, we will now add points D and E as maximal elements. To make sure that each element is comparable to exactly two other elements, we connect point C to both D and E. Now, points D and E are maximal elements, and both of them are comparable to exactly two other elements (points B and C for point D, and points A and C for point E). Diagram: D E | | C / \ A B
05

Verify the diagram

Finally, we should verify if the Hasse diagram meets all the conditions: 1. Two maximal elements: points D and E. 2. Two minimal elements: points A and B. 3. Each element is comparable to exactly two other elements: - A is comparable to B and C - B is comparable to A and C - C is comparable to A, B, D, and E - D is comparable to B and C - E is comparable to A and C We have successfully drawn a Hasse diagram that meets the given conditions.

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Key Concepts

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

Partially Ordered Set
A partially ordered set, often abbreviated as poset, is a key concept in order theory. It consists of a set coupled with a relation that defines the order among its elements. This relation is not just any relation, but one that fulfills specific conditions: reflexivity, antisymmetry, and transitivity. A poset differs from a totally ordered set (like rational numbers) because not all elements are necessarily comparable to each other.
  • Reflexivity ensures that every element relates to itself, giving each element a foundation in the order structure.
  • Antisymmetry restricts two different elements from being mutually dominant, maintaining a clear order with no ambiguity.
  • Transitivity ensures that the order propagates smoothly across elements, making multi-step connections possible.
In terms of visualization, a poset is often represented by a Hasse diagram, which is a simplified graph showing direct partial orders without redundant connections.
Maximal Elements
Within a partially ordered set, maximal elements are those that have no other elements above them when compared. These elements are "tops" in their local hierarchy because no greater element exists above them in the order criteria. Importantly, a poset can have multiple maximal elements, as seen in the example exercise, which included two maximal elements. When creating or analyzing a Hasse diagram, you would see maximal elements at the top of the diagram, with no edges leading outwards to any other elements. This visual cue is consistent with the idea that nothing exceeds a maximal element within that structure. Maximal elements do not necessarily mean they have an absolute maximum; rather, they simply have no successor in the poset.
Minimal Elements
Minimal elements in a partially ordered set are the opposite counterparts to maximal elements. They are at the base level regarding order, having no elements beneath them that are smaller. In other words, there exists no element in the poset that is smaller than the minimal elements by the order criteria. In the provided Hasse diagram, minimal elements are represented at the bottom, indicating their foundational standing in the hierarchy. Just like maximal elements, a poset can have multiple minimal elements. In the example, two minimal elements are featured, emphasizing that posets can be inclusive and diverse in how elements relate within their structure.
Reflexivity
Reflexivity is one of the three fundamental properties of a partial order relation. This property states that every element in the set is related to itself. Mathematically, for any element \( a \) in the set, \( aRa \) holds true, where \( R \) represents the relation.This property helps establish a basic grounding for every element, ensuring that they have a starting point in the ordering process. Reflexivity simplifies the relational structure by reinforcing that no element is isolated from itself, which could be useful especially in posets depicted by Hasse diagrams.
Antisymmetry
Antisymmetry is a foundational property of partial orders. This characteristic ensures that if one element is related to another and vice versa, then the two elements must be identical. Mathematically, for any elements \( a \) and \( b \), if \( aRb \) and \( bRa \), then \( a = b \).In a Hasse diagram, this principle can be seen when examining connections: the structure will not show loops or cycles that suggest different elements reciprocally dominating each other. By preventing such symmetric dominance, antisymmetry maintains clarity in the poset, distinguishing distinct elements robustly.
Transitivity
Transitivity forms another core aspect of the partial order relation. It ensures continuity of the relation within the poset. Simply put, if an element \( a \) is related to \( b \), and \( b \) is related to \( c \), then \( a \) must be related to \( c \). Mathematically, this is expressed as if \( aRb \) and \( bRc \), then \( aRc \).In practice, transitivity allows for seamless connections and helps deduce indirect order relations without needing explicit pairwise comparisons. When viewing a Hasse diagram, transitivity is implied in the multi-layer connections between elements: while only direct relationships are displayed, the transitive ones can be inferred, illustrating the coherent flow of relationships throughout the poset.

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Most popular questions from this chapter

Let \(A\) be the set of points in the rectangle with \(x\) and \(y\) coordinates between 0 and 1 . That is, $$ A=\\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid 0 \leq x \leq 1 \quad \text { and } \quad 0 \leq y \leq 1\\} $$ Define a relation \(R\) on \(A\) as follows: For all \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) in \(A_{1}\) $$ \begin{aligned} \left(x_{1}, y_{1}\right) R\left(x_{2}, y_{2}\right) \Leftrightarrow & \Leftrightarrow \\ \left(x_{1}, y_{1}\right)=\left(x_{2}, y_{2}\right) ; & \text { or } \\ x_{1}=0 & \text { and } x_{2}=1 \quad \text { and } \quad y_{1}=y_{2} ; \quad \text { or } \\ x_{1}=1 & \text { and } x_{2}=0 \quad \text { and } \quad y_{1}=y_{2} ; \quad \text { or } \\ y_{1}=0 & \text { and } y_{2}=1 \quad \text { and } \quad x_{1}=x_{2} ; \quad \text { or } \\ y_{1}=1 & \text { and } y_{2}=0 \quad \text { and } \quad x_{1}=x_{2} . \end{aligned} $$ In other words, all points along the top edge of the rectangle are related to the points along the bottom edge directly beneath them, and all points directly opposite each other along the left and right edges are related to each other. The points in the interior of the rectangle are not related to anything other than themselves. Then \(R\) is an equivalence relation on \(A\). Imagine gluing together all the points that are in the same equivalence class. Describe the resulting figure.

a. Find an inverse for 41 modulo 660 . b. Find the least positive solution for the following congruence: \(41 x \equiv 125(\bmod 660)\).

Define a relation \(R\) on the set of all real numbers \(\mathbf{R}\) as follows: For all \(x, y \in \mathbf{R}\), $$ x R y \quad \Leftrightarrow \quad x^{2} \leq y^{2} . $$ Is \(R\) a partial order relation? Prove or give a counterexample.

Suppose \(R\) and \(S\) are binary relations on a set \(A\). a. If \(R\) and \(S\) are reflexive, is \(R \cap S\) reflexive? Why? \(\boldsymbol{H}\) b. If \(R\) and \(S\) are symmetric, is \(R \cap S\) symmetric? Why? c. If \(R\) and \(S\) are transitive, is \(R \cap S\) transitive? Why?

Draw a Hasse diagram for a partially ordered set that has three maximal elements and three minimal elements and is such that each element is either greater than or less than exactly two other elements.
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