Question

Let \(A=(1,2,3,4\\} .\) Find the transitive closure of the relation \(R\) defined on \(A\) as $$R=\\{(1,2),(2,1),(2,3),(3,4)\\}$$

Short answer

The transitive closure of \(R\) is \((1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,4)\).

Step-by-step solution

Understand the Relation

Given the relation \( R = \{(1,2), (2,1), (2,3), (3,4)\} \), it is defined on set \( A = \{1, 2, 3, 4\} \). These pairs represent directed edges in a graph. Our task is to ensure that if there is a path from one element to another, there is a direct edge. This is what closure does in terms of reachability.

Determine Initial Paths

List out the paths formed by the relation: - Direct paths from the given pairs: - \(1 \rightarrow 2\), - \(2 \rightarrow 1\), - \(2 \rightarrow 3\), - \(3 \rightarrow 4\).

Find Paths of Length 2

Using the given pairs, construct new paths of length two:- From \(1\): 1. \(1 ightarrow 2 ightarrow 1\) gives \((1,1)\). 2. \(1 ightarrow 2 ightarrow 3\) gives \((1,3)\).- From \(2\): 1. \(2 ightarrow 1 ightarrow 2\) gives \((2,2)\). 2. \(2 ightarrow 3 ightarrow 4\) gives \((2,4)\).- From \(3\): 1. \(3 ightarrow 4\) does not extend the relation.- From \(4\): 1. No paths extend from 4 yet.

Find Paths of Longer Lengths if Needed

From step 3, we can further combine paths:- Check for paths starting from \(1\) and \(2\) that reach further: 1. \(1 ightarrow 3 ightarrow 4\) gives \((1,4)\).- Other attempts: Extending \(2\) to \(3\) to \(4\), such as \(2 ightarrow 4\), have already been captured or are nonexistent.

Compile the Transitive Closure

Collect all unique pairs from the original relation and the new pairs formed by transitivity:- Original relation: \((1,2), (2,1), (2,3), (3,4)\).- Paths leading to new pairs: \((1,1), (1,3), (1,4), (2,2), (2,4)\).- Total transitive closure of \(R\): \((1,2), (2,1), (2,3), (3,4), (1,1), (1,3), (1,4), (2,2), (2,4)\).

Key concepts

Relations in Set Theory

Understanding relations in set theory is crucial for exploring how elements in a set can be connected. A relation on a set is essentially a subset of the Cartesian product of the set with itself. For a set \( A = \{1, 2, 3, 4\} \), a relation \( R \) might consist of ordered pairs like \((1,2)\). This means element 1 is related to element 2.

Relations have specific properties that deeply influence their structure, such as reflexivity, symmetry, and transitivity. In our example, the task was to find the transitive closure of \( R \). This involves adding more pairs to \( R \) so that if there's a path from one element to another, a direct relation exists.

By extending this relation with additional pairs, we ensure maximal reachability among elements while keeping the structural logic intact. The key takeaway is that set theory provides a foundation to examine complex connections simply and methodically.

Directed Graphs

Directed graphs, or digraphs, help visualize relations in set theory. They consist of nodes connected by directed edges. Each edge has a direction, going from one node to another, just as an ordered pair \((x, y)\) means "from \(x\) to \(y\)."

In the given exercise, each pair in the relation represents an arrow pointing from one element to another, forming paths on a graph. For example, the pair \((2,3)\) forms a directed edge from node 2 to node 3. These edges show possible paths for movement or transfer between nodes, highlighting how one element can connect with another within the system.

Directed graphs are particularly useful for understanding problems involving reachability and connectivity. By examining directed paths, you can reveal how elements in a set interact, ultimately constructing a visual framework of the transitive closure.

Graph Reachability

Graph reachability is about determining if there is a path from one node to another in a graph. Reflecting on our example, given the elements \(1\) to \(4\), reachability dives into understanding if one element can "reach" another element through a defined path.

In creating a transitive closure, we investigate all possible paths, direct and indirect, that connect nodes. For instance, while \((2,3)\) directly connects elements, additional paths like \((1,2)\) followed by \((2,3)\) imply \((1,3)\), extending the connection further through intermediate nodes.

Finding the transitive closure is about capturing these indirect paths, cementing them as direct links between nodes. This accomplishment ensures that the graph completely expresses all potential reachabilities, transforming an initially partial framework into a comprehensive network of connections.