Question

Represent by a digraph the partial order divides defined on the integers 0 through \(11 .\)

Short answer

The digraph shows integers 1 to 11, with edges from \(a\) to \(b\) if \(a\) divides \(b\), simplified by removing transitive edges.

Step-by-step solution

Understanding Partial Orders

A partial order is a binary relation over a set that is reflexive, antisymmetric, and transitive. Here, the relation is 'divides' on the integers from 0 through 11, meaning we examine how numbers can divide each other without leaving a remainder.

Identifying the Set and Relation

We need to consider the set of integers from 1 to 11 (since 0 cannot be a divisor in this context). The relation 'divides' means if number A divides number B, then there is a directed edge from A to B in the digraph.

Constructing the Divisibility Relationships

List all pairs \((a, b)\) such that \(a \leq b\) and \(a\) divides \(b\). For example, 1 divides every number, and 2 divides 2, 4, 6, 8, 10, etc.

Building the Digraph

Create a digraph with vertices representing each integer from 1 through 11. For directed edges, connect \(a\) to \(b\) if \(a\) divides \(b\).

Simplifying the Digraph with Transitivity

Remove redundant edges by applying transitivity. If there is a path from A to B through another vertex C, then A to B does not need a direct edge. This simplifies the diagram and maintains the partial order property.

Key concepts

Binary Relation

In mathematics, a binary relation is a crucial concept that connects elements of one set to elements of the same or another set. A binary relation on a set contains ordered pairs \(a, b\), where \(a\) and \(b\) are elements of the set. This relation describes how one element relates to another. Binary relations can have various properties that help us understand their behavior and applications.

• Example: If a binary relation "divides" exists on the set of integers from 1 to 11, then the relation consists of pairs such as \( (2, 4)\), \( (3, 6)\), indicating that 2 divides 4 and 3 divides 6.

Binary relations are foundational for constructing more complex relations, such as partial orders.

Reflexive

A reflexive relation is one where every element is related to itself. In other words, for a set \(A\), the reflexive property means that for all \(a \in A\), the pair \( (a, a)\) belongs to the relation. Reflexive relations ensure that each element has the opportunity to "divide" itself, especially in divisibility contexts.

• Example: In the set of integers from 1 to 11, reflexivity in the divides relation indicates that each integer divides itself, i.e., \( (1, 1)\), \( (2, 2)\), \( (3, 3)\), etc.

Reflexivity is a stepping stone in establishing other properties like antisymmetry and transitivity essential for partial orders.

Antisymmetric

Antisymmetry is a property of a binary relation where for any two elements \(a\) and \(b\) in a set, if both \( (a, b)\) and \( (b, a)\) are in the relation, then \(a\) must be equal to \(b\). This means that if \(a\) and \(b\) are distinct, one cannot divide the other in the opposite pairing.

• Example: If 3 divides 6, then in an antisymmetric relation, 6 cannot divide 3. Thus, only the pair \( (3, 6)\) exists, maintaining antisymmetry.

Antisymmetry is significant for partial orders as it enforces a clear direction of division among elements of the set.

Transitive

Transitivity is a property that allows simplification of relations by deducing new information from existing relations. In a transitive relation, if \(a\) relates to \(b\) and \(b\) relates to \(c\), then \(a\) must relate to \(c\). This property eliminates redundant relations by recognizing indirect connections.

• Example: For integers, if 2 divides 4 and 4 divides 8, transitivity implies that 2 divides 8 as well, eliminating the need for a separate edge from 2 to 8 if there is already a path through 4.

Transitivity greatly helps in constructing simplified and efficient representations of relations like digraphs in the study of partial orders.