Question

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

Short answer

The minimal elements are 1, 7, and 11; the maximal elements include 7 and 11; the minimum element is 1. There is no maximum element.

Step-by-step solution

Understand 'Divides' Partial Order

The set is \(\{1,2,3,4,5,6,7,8,9,10,11\}\). In this context, 'divides' means that for any two elements \(a, b\) in the set, \(a \mid b\) if there exists an integer \(k\) such that \(b = ak\). This order relation is reflexive, antisymmetric, and transitive.

Determine Pairs with Divisibility

For each element \(x\) in the set, identify other elements \(y\) such that \(x \mid y\). E.g., \(1\) divides every element, \(2\) divides 2, 4, 6, 8, and 10; \(3\) divides 3, 6, and 9, etc. Perform this check for all elements in the set.

Draw the Hasse Diagram

Construct a Hasse diagram where each element is a node. Use edges to connect nodes if one element divides another directly (with no intermediate division by another element). Begin with 1 at the bottom, since it divides every number, and arrange other nodes upwards based on divisibility.

Identify Minimal and Minimal Elements

In a poset, minimal elements are those that are not divisible by any smaller number (except 1). Here, the minimal elements are 1 (which divides all), and 7, 11 (which are prime in this set and not divisible by others).

Identify Minimal and Minimum Elements

The minimum element is the smallest element that can divide all others in the set. Here, the minimum element is \(1\) because it divides every element in the set.

Identify Maximal Elements

Maximal elements are those that are not divided by any larger numbers in the set. In this set, each number is maximal with respect to itself, meaning that they appear at the top of the Hasse diagram and no larger element divides them directly. Here, the maximal elements are 7, 11, and perhaps a few others at the top of the diagram depending on specific pairs checked.

Identify Maximum Elements

A maximum element is one that is above all others in terms of divisibility. In this digraph, no single number divides all others (except by 1, which is at the bottom in this case). Therefore, there is not a clear maximum element above all others.

Key concepts

Divisibility

Divisibility is a key concept in mathematics, especially when discussing sets and order relations. It refers to the ability of one number to be divided by another with no remainder. In our context, we explore divisibility within a set of integers \( \{1,2,3,4,5,6,7,8,9,10,11\} \). For any two elements \( a \, b \) from this set, \( a | b \) implies there exists an integer \( k \) such that \( b = ak \.\) Thus, divisibility defines the relationship and order between numbers.

• Reflexive: Every number divides itself. Hence, \( a | a \).
• Antisymmetric: If \( a | b \) and \( b | a \,\) then \( a = b\).
• Transitive: If \( a | b \) and \( b | c \,\) then \( a | c \).

By applying these properties, we can identify relationships among the set elements, which can visually represent the order structure using diagrams like the Hasse diagram.

Hasse Diagram

A Hasse diagram is a graphical representation of a partial order. It simplifies understanding partial orders by showing elements and their direct relationships. We use it here to illustrate the divisibility among elements in our given set. In a Hasse diagram:

• Nodes: Represent the elements of the set.
• Edges: Connect nodes to indicate direct divisibility without any intermediate steps.
• No arrows: Though the diagram has direction (from lower elements to higher), arrows are not typically drawn.

To draw a Hasse diagram for our set, start with 1 at the bottom since it divides all other numbers. Other numbers are arranged above 1 based on divisibility. For instance, 2 connects directly to 4, 6, 8, and 10, forming a chain of divisibility. This visualization allows us to grasp the hierarchical structure of the set in terms of divisibility, making it easier to identify elements with specific properties.

Minimal Elements

Minimal elements in a partially ordered set are those that have no smaller element dividing them, apart from the reflexive property of divisibility by 1. Therefore, they cannot be ordered lower by divisibility from another element, except 1. In our specific set \( \{1,2,3,4,5,6,7,8,9,10,11\} \,\) the minimal elements include:

• 1: Divides all numbers and hence stands alone as a base reference.
• 7 and 11: These are prime numbers within the set, meaning they are not divisible by any other number besides 1.

Understanding minimal elements helps in recognizing the foundational building blocks within a set, laying out which elements serve as basic constituents under the divisible order structure.

Maximal Elements

Maximal elements in a partially ordered set are those which are not divided by any larger element within that set. In simpler terms, no other element follows them directly in terms of divisibility. In our set \( \{1,2,3,4,5,6,7,8,9,10,11\} \,\) identifying maximal elements helps depict the end nodes or peaks in the Hasse diagram:

• 7 and 11: Since these are prime numbers, they receive no division from other elements within this set.
• Others: Certain other numbers may also be classified as maximal based on their positional relationships within the diagram, such as 9 or 5, as no higher numbers within this set divide them.

Maximal elements help visualize where certain chains of divisibility conclude within the ordered structure, crucially shaping the diagram's overall form.