Question

Let \(R\) be the relation on {1,2,3,4,5} with elements \(\\{(1,1),(2,1),(3,2),(2,3),(1,4)\). (3,5),(5,2)} . Represent the symmetric closure of \(R\) as a digraph.

Short answer

The symmetric closure digraph has bi-directional edges for symmetric pairs.

Step-by-step solution

Understand the Relation and Its Elements

The given relation \( R \) consists of the ordered pairs: \((1,1), (2,1), (3,2), (2,3), (1,4), (3,5), (5,2)\). This means that there is a directed edge from \(a\) to \(b\) for each pair \((a,b)\). We will use this as our starting point to create a symmetric closure.

Identify Missing Symmetric Pairs

For symmetric closure, if \((a, b)\) is in \(R\), then \((b, a)\) should also be in \(R\). Check each pair to see if its reverse is present in \(R\). Reverse pairs not in \(R\) need to be added: \((1, 2), (2, 3), (5, 3)\).

Add Missing Symmetric Pairs to R

Add the identified symmetric pairs to make \(R\) symmetric: Include \((1, 2), (3, 2), (2, 5)\) to \(R\). The symmetric closure, \(R_s\), consists of: \((1,1), (2,1), (1,2), (3,2), (2,3), (3,5), (5,2), (5,3), (2,5), (1,4)\).

Draw the Digraph

Represent each element as a vertex in the digraph. Draw directed edges according to the pairs in the symmetric closure \(R_s\). There should be bi-directional arrows (or undirected edges) where symmetric pairs exist, such as between nodes 2 and 3.

Key concepts

Discrete Mathematics

Discrete Mathematics is a branch of mathematics dealing with structures that are distinct and separate. While it might seem a little abstract, it plays a crucial role in computer science and related fields. The concept of discrete structures includes things like integers, graphs, and statements in logic. It's the mathematics of countable, distinct elements. Discrete Mathematics covers topics such as:

• Graph Theory: The study of graphs and networks, which are used to model pairwise relations among objects.
• Combinatorics: The science of counting, organizing, and understanding the structure of objects.
• Set Theory: The foundation of most mathematical disciplines, dealing with the collection of objects.
• Logic: Used in computer science to tackle decision-making problems.

These areas form the building blocks for more complex structures in mathematics. Discrete Mathematics is particularly well-suited for computer algorithms, cryptography, and implementing software solutions.
It provides the mathematical basis for reasoning and decision making in software development and computational problems.

Relations and Digraphs

Relations in Discrete Mathematics describe how elements from one set relate to elements of another set or the same set. A relation from set A to set B is a subset of the Cartesian product, A x B. Establishing a relation typically involves looking at which elements from set A connect to which elements in set B.

Relating to this is the concept of digraphs or directed graphs, which serve as a visual way to represent relations. A digraph involves:

• Vertices (or nodes): Representing the elements of a set.
• Directed edges (or arcs): Depicting a relationship from one vertex to another.

If there's a relation from one element to another, this is shown by a directed edge from one node to another. Creating digraphs helps in visualizing complex relationships and finding out how elements interconnect or communicate with each other.
In our exercise, the relation's elements are represented as ordered pairs in a set, depicted as vertices and directed edges in a digraph. Relationships in a symmetric closure are drawn with bidirectional arrows, indicating that if a path exists from A to B, then there is also a path from B to A.

Symmetric Relations

Symmetric relations are a specific type of relation in Discrete Mathematics where if one element is related to another, then the reverse is always true. In other words, for a relation to be symmetric, whenever there is a pair \(a, b\), there should also be a pair \(b, a\) included in the relation.

This concept is crucial when dealing with networks or connections that are reciprocal, meaning information or some entity can flow in both directions equally. To create a symmetric relation from a non-symmetric one, we perform something called a symmetric closure. This process involves:

• Checking each element pair to see if the inverse pair is present.
• Adding the missing inverse pairs that aren't already included.

Taking symmetric closure guarantees that all relationships are mutual. In graphical terms, it means converting directed edges into bidirectional or two-way edges.
This concept is vital in various fields like network theory, where mutual connections are significant; ensuring data moves not just from source to destination but back in the data stream. In our exercise, we found missing symmetric pairs and added them to make the relation symmetric, visualized using a digraph, where elements form two-way, directed edges between connected nodes.