Problem 6 - Applied Discrete Structures

Question

Let \(A=\\{a, b, c, d\\}\). Let \(r\) be the relation on \(A\) with adjacency matrix \(\begin{array}{llll}a & b & c & d\end{array}\) $$ \begin{array}{l} a \\ b \\ c \\ d \end{array}\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right) $$ (a) Explain why \(r\) is a partial ordering on \(A\). (b) Draw its Hasse diagram.

Step-by-Step Solution

Answer

Relation is a partial order because it is reflexive, antisymmetric, and transitive. The Hasse diagram shows b between c and d, with a being less than b.

Step 1: Understanding Partial Order

A relation is a partial order if it is reflexive, antisymmetric, and transitive. We'll check these properties for the given relation on set \( A = \{a, b, c, d\} \).

Step 2: Check Reflexivity

The relation is reflexive if every element is related to itself. Looking at the diagonal of the matrix (\(1, 1, 1, 1\)), all entries are 1, confirming that every element in \( A \) is related to itself. Thus, the relation is reflexive.

Step 3: Check Antisymmetry

A relation is antisymmetric if, for any two elements \( x \) and \( y \), if \( xRy \) and \( yRx \), then \( x = y \). From the matrix, there are no pairs (except for the ones on the diagonal where \( x = y \)) where both \( xRy \) and \( yRx \) are 1. Hence, the relation is antisymmetric.

Step 4: Check Transitivity

The relation is transitive if whenever \( xRy \) and \( yRz \), then \( xRz \) must also hold. For element 'c', since we have \( cRc\, cRb\, (and)\, bRa\), we have a transitive chain leading to \( cRa \). All other possible chains are closed; therefore, the relation is transitive.

Step 5: Drawing the Hasse Diagram

The Hasse diagram omits reflexive loops and shows direct relations without transitivity edges. From the matrix, we see the relations (ignoring self-loops) as \( a \), \( b \rightarrow b \),\( c \rightarrow a \), \( c \rightarrow b \), \( c \rightarrow c \), \( d \rightarrow b \), and \( d \rightarrow d \). Thus the Hasse diagram is: a lower element joining to b, and c joining down to both a and b, while d joins to b.

Key Concepts

Reflexivity

Reflexivity is a property in relation where each element must relate to itself. Imagine looking at yourself in a mirror; reflexivity in a relation works similarly. For a relation to be reflexive, every element in the set must have a loop back to itself.

In mathematical terms, if we have a set \(A = \{a, b, c, d\}\), a relation \(r\) is reflexive if for every element \(x\) in \(A\), the pair \((x, x)\) is in the relation \(r\). Viewing it through a matrix, this simply means that all the entries on the diagonal should be 1.

In our example using the given adjacency matrix, if we check the diagonal from top left to bottom right, we find 1s. Thus, each element \(a, b, c, \text{and } d\) is indeed related to itself. That confirms reflexivity for this relation. This is an essential stepping-stone in determining whether a relation can be considered a partial order.

Antisymmetry

Antisymmetry is another property vital for establishing a partial order. It's a condition that ensures order by stating that if two elements relate to each other mutually, they must be identical.

Here's how it works: in a relation \(r\) on a set \(A\), if the pairs \((x, y)\) and \((y, x)\) are both in \(r\), then \(x\) must be equal to \(y\). This means that if two elements are linked in both directions, they cannot be distinct.

In our given relation, we examined the pairs in the matrix and confirmed there are no two different elements that have links both ways, except for the diagonal where \(x = y\). Thus, this relation satisfies antisymmetry because it complies with the rule of mutual relation leading to equality. Antisymmetry helps in simplifying the relationship visualization in a set, paving the way for understanding more complex structures like lattices.

Transitivity

Transitivity is akin to a chain of dominoes where the fall of one triggers the next. For a relation to be transitive, whenever an element \(x\) relates to \(y\), and \(y\) relates to \(z\), then \(x\) must relate to \(z\). This is like saying if you have a direct route from \(x\) to \(y\) and \(y\) to \(z\), you also have a path from \(x\) to \(z\).

In our exercise, examining the matrix and its structure, every indirect connection works through this domino principle. For instance, since \(c\) relates to \(b\) and \(b\) relates to \(a\), it follows that \(c\) also relates to \(a\). Each element in the matrix upholds this rule, affirming that the relation is indeed transitive. Transitivity allows for a streamlined representation of complex relationships, as seen in tasks like drawing Hasse diagrams.

Hasse Diagram

A Hasse diagram is a simple, intuitive tool for visualizing a partially ordered set. Think of it as a map that showcases the connections between elements without clutter.

Constructing a Hasse diagram involves some straightforward steps:

• Start by listing all elements.
• Remove any reflexive loops, as they are implied.
• Simplify by omitting transitive edges, focusing on direct relationships.

For our specific exercise, following these guidelines led to a neat visualization. The relation from \(c\) to \(a\) and \(b\), \(b\) being related to itself, and so forth, is depicted clearly. The simplicity of the Hasse diagram lies in its elegant representation of direct connects, highlighted without extraneous details.

Using a Hasse diagram can vastly improve one's understanding of how each component of a set leads to or connects with others, offering insights into the entire relational model.