Question

(a) Draw a diagram to represent the \(\mid\) (divides) partial order on \(\\{0,1,2,3,4,5,6,7\), \(8,9,10,11 \mid\) (b) Identify all minimal, minimum, maximal, and maximum elements in the diagram.

Short answer

Minimal elements: 1, 3, 5, 7, 9, 11; Maximal elements: 8, 9, 10, 11. No single maximum element. Minimum elements include 1 and prime numbers.

Step-by-step solution

Understand the Set and Relationship

We are given a set of elements \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\) with the partial order defined by the divisibility relation "|". This means that for any elements \(a\) and \(b\) in the set, \(a \mid b\) if \(b\) can be divided evenly by \(a\).

Construct the Hasse Diagram

Start by identifying pairs \((a, b)\) where \(a \mid b\) and \(a eq b\). Exclude the trivial "\(a \mid a\)" connections. For example, \(1\) divides all numbers, \(2\) divides 2, 4, 6, 8, 10, \(3\) divides 3, 6, 9, etc. Construct a Hasse Diagram omitting transitive edges and self-loops, representing elements as nodes and relations as lines connecting them.

Identify Minimal Elements

Minimal elements are those that have no other element dividing them apart from the element itself. In this set, the number \(0\) can be ignored since it divides every number. Therefore, minimal elements are those that are only at the bottom of the Hasse diagram arrows sequence. Here, the minimal elements are \(1\) and prime numbers (\(1, 3, 5, 7, 9, 11\)).

Identify Maximum Element

The maximum element is the element at the top of the Hasse diagram, which means it is divisible by others but divides none further up the diagram. Here, the number \(0\) is an exceptional maximal element (but divisibility technically does not work with \(0\)), but within non-zero numbers, there is no single number that divides all others, so no non-zero maximum exists in the set \(\{1, 2, \, \dots, 11\}\).

Identify Minimal Elements Correctly

Correction from Step 3, noting non-prime minimal numbers: Minimals are exact nodes that are not divisible by others in the list. They are: \(1, 2, 3, 5, 7, 9, 11\).

Identify Minimal and Maximal Elements

Minimal elements are \(1, 2, 3, 5, 7, 9, 11\) since they are not divisible by any other smaller element apart from themselves. Maximal elements are those which do not have any other elements above them in the diagram -- these include the largest non-divisible elements by others: \(8, 9, 10, 11\). There is no single maximum common across the entire set inside positive numbers.

Key concepts

Hasse Diagram

A Hasse diagram provides a visual representation of a partial order. In our example, the set \( \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \} \) is organized using the divisibility relation "|", meaning \( a \mid b \) if \( b \) is divisible by \( a \) without a remainder. The diagram is constructed by identifying elements and connecting them with edges to show this relation.

To create the Hasse diagram, begin by listing all divisibility relationships. Start from the smallest elements and find what they divide. You'll notice that, for instance, \( 1 \) divides every number in the set since any number divided by 1 is itself. Remove the reflexive \( a \mid a \) and transitive edges to keep the diagram simple and clear. Each line or edge in the diagram represents a direct divisibility link between two elements. These edges rise upwards from a lower number (divisor) to a higher number (dividend). By omitting transitive links (higher-level links between indirectly connected elements), the Hasse diagram remains neat and interpretable.

Divisibility Relation

The divisibility relation "|" is a foundational concept in discrete mathematics, particularly in set theory and order relations. In a set \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \), the divisibility relation \( a \mid b \) means that \( b \) can be divided by \( a \) without leaving a remainder. This forms a partial order because it satisfies three main properties: reflexivity, antisymmetry, and transitivity.

• Reflexivity ensures that every element divides itself, meaning \( a \mid a \).
• Antisymmetry dictates that if \( a \mid b \) and \( b \mid a \), then \( a = b \).
• Transitivity allows for successive applications, where if \( a \mid b \) and \( b \mid c \), then \( a \mid c \).

In our divisibility set, numbers like 1 have the property of dividing every number, natural primes such as 5 or 7 only divide themselves and the numbers that directly follow in certain sequences, reflecting their minimal status.

Minimal and Maximal Elements

In any partially ordered set, minimal and maximal elements play key roles. A minimal element is any element that is not divisible by any other element except itself. On a Hasse diagram, they appear as starting points or lowest nodes. In our set, the minimal elements are \(1, 2, 3, 5, 7, 9,\) and \(11\), each dividing no numbers smaller than themselves except for themselves.

Conversely, maximal elements are those that no other element divides further in the diagram. These are top nodes where lines do not extend further upwards. In this set under the divisibility partial order, the maximal elements are \(8, 9, 10,\) and \(11\). However, no number in the non-zero set of \( \{1, 2, \ldots, 11\} \) universally divides all the others, implying the absence of a single maximum element across that set.

Interestingly, 0 can be considered an intriguing case because it technically divides every number, but divisibility by 0 is typically undefined in mathematical contexts.

Discrete Mathematics

Discrete mathematics involves the study of mathematical structures that are fundamentally countable or distinct. Unlike continuous mathematics, which involves topics in calculus and seamless transitions, discrete math focuses on topics such as integers, graphs, and logical statements, where objects can be distinctly separated.

One of the essential uses of discrete mathematics is its applications in computer science, where binary logic and algorithms are pivotal. In contexts like our exercise, discrete math principles help define relationships, such as the divisibility relation, and their representations through graphs like Hasse diagrams.

It introduces concepts of order and hierarchy within sets, often solving problems of enumeration, combination, and connectivity. Understanding partial orders, as seen in this exercise, is crucial for grasping how elements interact in logically defined structures. Overall, discrete mathematics forms the backbone of subjects that require clear distinctions between discrete entities.