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 reflexive closure of \(R\) as a digraph.

Short answer

Add self-loops at 2, 3, 4, and 5 to make the relation reflexive, then draw the directed graph.

Step-by-step solution

Understanding Reflexive Closure

The reflexive closure of a relation on a set is obtained by adding the minimal number of ordered pairs to make the relation reflexive. A relation is reflexive if every element is related to itself.

Identifying Missing Elements for Reflexivity

The given set is \((1,2,3,4,5)\), and relation \(R\) includes \((1,1),(2,1),(3,2),(2,3),(1,4),(3,5),(5,2)\). To make \(R\) reflexive, we must ensure that \((1,1), (2,2), (3,3), (4,4),\) and \((5,5)\) are in the relation, since these cover all elements relating to themselves.

Adding Missing Pairs

In the given relation, we already have \((1,1)\). We need to add \((2,2), (3,3), (4,4), (5,5)\) to the relation to make it reflexive.

Final Set of Relations

The final reflexive closure of \(R\) consists of the elements:\((1,1), (2,1), (3,2), (2,3), (1,4), (3,5), (5,2), (2,2), (3,3), (4,4), (5,5)\).

Drawing the Digraph

To represent this relation as a digraph, draw a directed graph with vertices 1, 2, 3, 4, and 5. Draw directed edges according to the pairings in the reflexive closure. For example, an edge from vertex 2 to vertex 1 for the pair \((2,1)\), and loops at vertices 2, 3, 4, and 5 for the pairs \((2,2), (3,3), (4,4)\), and \((5,5)\).

Key concepts

Relations in Discrete Mathematics

In discrete mathematics, the concept of a relation is vital for understanding how different elements within a set interact or connect with one another. A relation essentially describes a relationship between elements of one or more sets. When we talk about a relation on a single set, we are focusing on how elements from that set are paired together. For instance, if we have a set \(A = \{a, b, c\}\), a relation on this set might be \(R = \{(a,a), (b,a), (c,c)\}\). Here, the ordered pairs show how elements are related.

A good way to visualize relations is by thinking of them as connections or links between nodes in a network. In mathematics, these nodes are the elements, and the connections are the relationships. Relations can be described as:

• Reflexive: An element is related to itself.
• Symmetric: If an element is related to another, the second is related to the first.
• Transitive: If an element is related to a second element, which in turn is related to a third, the first element is related to the third.

Understanding how relations work and what properties they have is crucial as it forms the backbone for more complex mathematical concepts, especially in fields like graph theory and relational databases.

Directed Graphs

Directed graphs, often referred to as digraphs, are a way of visually representing relations. In a directed graph, elements of a set are represented as vertices (or nodes), and the relations between them are represented as directed edges (or arrows). Each edge has a direction, indicating a one-way relationship from one vertex to another.

For example, if a set has elements \(\{1, 2, 3\}\) and the relation \(\{(1,2), (2,3)\}\), we would draw an arrow from vertex 1 to vertex 2, and another arrow from vertex 2 to vertex 3. This creates a visual map of the relation. Each arrow points from the first element of the ordered pair to the second.

Directed graphs are powerful because of their ability to make complex relationships clearer, helping students and professionals alike to identify patterns or properties like cycles and connectivity. Reflexive relations are depicted by loops—arrows that start and end at the same vertex, indicating that an element is related to itself.

Reflexivity in Mathematics

Reflexivity is a fundamental property in relation theory, where every element in a set is related to itself. In terms of mathematical relations, a set is said to be reflexive if for every element \(a\) in set \(A\), the pair \( (a, a) \) exists within the relation.

When we talk about reflexive closure, we mean altering a non-reflexive relation to make it reflexive by adding a minimal number of pairs. For example, consider the set \(A = \{1, 2, 3\}\) with a relation \(R = \{(1, 2), (2, 3)\}\). To make this relation reflexive, we would add \( (1, 1), (2, 2), (3, 3) \) to the set, so all elements relate to themselves as well as to other elements.

Understanding reflexivity helps in many areas, like ensuring network connectivity, where each node should at least connect to itself to ensure complete access. By grasping how reflexivity functions in a set and its graphical representation, learners can apply these principles effectively to solve problems involving complex relations.