Question

Let \(R\) be the relation on (1,2,3,4} with elements {(1,1),(2,1),(3,2),(2,3),(1,4)} . Represent the transitive closure \(R^{*}\) of \(R\) as a digraph.

Short answer

Draw the digraph for \(R^{*}\) with edges: \{(1,1), (2,1), (3,2), (2,3), (1,4), (2,4), (3,4), (3,1)\}.

Step-by-step solution

Understand the Elements in R

The relation \( R \) consists of the elements \{(1,1), (2,1), (3,2), (2,3), (1,4)\}. Each element \((a, b)\) indicates a directed edge from vertex \(a\) to vertex \(b\) in the digraph.

Initial Graph Representation

Start by drawing vertices for each element in the set \{1, 2, 3, 4\}. Add directed edges for each pair present in the relation \( R \). This serves as the initial digraph representation of \( R \). Connect 1 to 1, 2 to 1, 3 to 2, 2 to 3, and 1 to 4.

Adding Transitive Arcs

A relation is transitive if whenever there is an edge from \(a\) to \(b\) and from \(b\) to \(c\), there must be a direct edge from \(a\) to \(c\). Examine all possible pairs in the current digraph and add the necessary direct edges to ensure transitivity. For instance, since there's a path from \(2\) to \(1\) and from \(1\) to \(4\), add a direct edge from \(2\) to \(4\). Also, since there's a path from \(3\) through \(2\) to \(1\), add a direct edge from \(3\) to \(1\). Similarly, since there's a path from \(3\) through \(2\) to \(4\), add a direct edge from \(3\) to \(4\).

Verification

Verify that the graph is now transitive. Check that for any pair of edges \((a, b)\) and \((b, c)\), there is a direct edge \((a, c)\). At this point, the graph of \( R^{*} \) should have these edges: \{(1,1), (2,1), (3,2), (2,3), (1,4), (2,4), (3,4), (3,1)\}, ensuring all transitive relationships are represented.

Draw Final Transitive Closure Digraph

With all transitive edges added, draw the final digraph. This graph is \( R^{*} \): the transitive closure of \( R \). The vertices \{1, 2, 3, 4\} should now include directed edges reflecting both the original relation and the transitive additions.

Key concepts

Relation

In the realm of discrete mathematics, a **relation** is a way of showing how elements from one set may be associated with elements of another set. An important characteristic of relations is their ability to define connections in a structured way. For example, consider a set \( \{1, 2, 3, 4\} \) where a relation \( R \) is described by pairs such as \((1,1), (2,1), (3,2), (2,3), (1,4)\). Each pair indicates a connection between elements. The first element of the pair points to the second, much like saying "1 is related to 4."
To consider the structure of relations more deeply, it's helpful to explore different kinds of properties such as:

• Reflexivity: Every element is related to itself.
• Symmetry: If an element \(a\) is related to \(b\), then \(b\) is related to \(a\).
• Transitivity: If an element \(a\) is related to \(b\) and \(b\) is related to \(c\), then \(a\) must be related to \(c\).

In this problem, our focus is on transitivity as we attempt to create a complete picture of all possible connections given the initial relations.

Digraph

A **digraph**, or directed graph, is a representation that uses directed edges to show the relationships between objects. In a digraph, nodes represent the objects, and arrows or directed lines illustrate the relationships or connections between these nodes.
In our specific example, the set \( \{1, 2, 3, 4\} \) can be represented by nodes, and each pair such as \( (2, 3) \) in the relation translates into a directed arrow from node 2 to node 3. This visual structure makes it easier to grasp how different elements relate to each other within a set.
Constructing this starts by:

• Drawing vertices for each element of the set.
• Adding directed edges for each pair in the relation.

Once the initial digraph is created, the task is to ensure that the graph reflects all possible transitive connections by adding any necessary additional edges. This results in a transformation from a simple digraph to its transitive closure.

Discrete Mathematics

**Discrete mathematics** is a fascinating field that deals with distinct and separate values. Unlike areas that focus on continuous values, such as calculus, discrete mathematics centers on countable, often finite sets. This makes it particularly powerful for computer science, logic, and practical problem-solving.
When working with relations and graphs, discrete mathematics gives us tools to model and solve problems involving finite systems. These models can include:

• Graph Theory: The study of graphs and digraphs used to solve problems based on connectivity, flow, and networks.
• Set Theory: Understanding and manipulating collections of elements.
• Combinatorics: Counting, arrangement, and combination of sets.

In the context of our exercise, discrete mathematics provides the framework for understanding relations, working with digraphs, and applying concepts like the transitive closure, ensuring we have explored all levels of possible connectivity in a graph.

Graph Theory

**Graph theory** is a vital branch of mathematics concerned with the study of graphs, which are mathematical structures used to model pairwise relations between objects. It focuses on the properties and applications of these structures.
Graphs in graph theory can be either directed (digraphs) or undirected, and they find applications in numerous fields, from sociology to computer algorithms. Graphs are useful when considering problems of connectivity, finding shortest paths, or modeling complex networks.
For the problem we are tackling, graph theory provides us with the language and toolkit to visualize and analyze the relation \( R \). By considering \( R \) as a graph, we can apply graph theory concepts like paths and connectivity. Building on this, creating a graph’s transitive closure means ensuring that all indirect connections are represented with direct edges. This step-by-step examination lays the foundation not only for understanding specific problems but also for larger applications in data structures and network analysis.