Question

Let \(R\) be the relation on \(\mid a, b, c, d, e, f, g\\}\) defined as $$R=\\{(a, b),(b, c),(c, a),(d, e),(e, f),(f, g)\\}$$ Find the smallest integers \(m\) and \(n\) such that \(0

Short answer

m = 3, n = 4. Transitive closure adds cycles, reflexive adds identity pairs, and symmetric adds reverse pairs.

Step-by-step solution

Understanding the Relation

The given relation \(R\) is defined on the set \(\{a, b, c, d, e, f, g\}\) with elements \((a, b), (b, c), (c, a), (d, e), (e, f), (f, g)\). Each pair \((x, y)\) indicates a direction from \(x\) to \(y\). We will analyze \(R\) to find its closures and powers.

Computing Powers of R

We compute \(R^2\) by finding pairs \((x, z)\) such that there exists a middle element \(y\) where both \((x, y)\) and \((y, z)\) are in \(R\). Starting with \((a, b)\) and \((b, c)\) yields \((a, c)\), and \((b, c)\) and \((c, a)\) yield \((b, a)\), and so forth.

Continue Evaluation Until Stability

Compute the third power, \(R^3\), which gives new pairs using the transitive steps through \((a, c)\), \((b, a)\), and so on. Continue until the powers stabilize such that no new pairs are generated.

Determine m and n for Stabilization

Rough calculations: After repeating the above steps, we find that \(R^3\) and \(R^4\) stabilize at equivalent sets of pairs, showing \(m = 3\) and \(n = 4\) as they give equal power relations.

Find the Transitive Closure

The transitive closure \(R^+\) adds pairs to make the set transitive: \((a, c), (b, a), (c, b)\). Also see \((d, f), (d, g), (e, g)\) due to chaining to the original pairs.

Calculate Reflexive and Symmetric Closures

The reflexive closure adds all reflexive pairs like \((a, a), (b, b)\) through to \((g, g)\). The symmetric closure involves adding reverse pairs for each pair in \(R\), such as \((b, a)\) to complement \((a, b)\), and so forth.

Key concepts

Relation

A **relation** on a set is a collection of ordered pairs consisting of elements from the set. In our original exercise, the relation \(R\) is defined on the set \(\{a, b, c, d, e, f, g\}\), and includes pairs such as \((a, b)\) and \((b, c)\). Each pair demonstrates a directed connection from one element to another. Relations are fundamental in mathematics and computer science because they help describe how elements relate to one another. A relation can be part of various types of closures, which are methods to extend the relation's properties to make it more comprehensive or satisfy specific conditions.

Reflexive Closure

The **reflexive closure** of a relation is created by adding pairs to a relation to ensure that it contains a loop at each element. That is, every element must relate to itself, which means for any element \(x\) in the set, the pair \((x, x)\) should exist in the relation.This is achieved by supplementing the original relation with these pairs:

• \((a, a)\)
• \((b, b)\)
• \((c, c)\)
• \((d, d)\)
• \((e, e)\)
• \((f, f)\)
• \((g, g)\)

The reflexive closure guarantees that every element includes a self-loop, ensuring it is reflexive.

Symmetric Closure

The **symmetric closure** of a relation is formed by ensuring that for every pair \((x, y)\) in the relation, its reverse \((y, x)\) is also included. This characteristic aims to make the relationship two-way. For instance, if our relation has the pair \((a, b)\), we add \((b, a)\) to the symmetric closure if it wasn't already present. With this logic applied to all elements of the set, symmetric closure of the original relation \(R\) involves:

• For \((a, b)\), add \((b, a)\)
• For \((b, c)\), add \((c, b)\)
• For \((c, a)\), add \((a, c)\)
• And similarly for pairs involving elements \(d, e, f, g\)

By completing this process, the relation becomes symmetrical.

Powers of a Relation

The **powers of a relation** involve composing a relation with itself multiple times. When a relation \(R\) is defined, one can compute its second power \(R^2\), third power \(R^3\), and so on. This involves forming new pairs by following paths through intermediary elements.Consider \(R^2\) of our example, calculated by examining sequences like \((a, b)\) and \((b, c)\) to produce the new pair \((a, c)\). Continue this chaining process through the sequence:

• \((a, c)\)
• \((b, a)\)
• \((c, b)\)

By repeatedly applying this process, you evaluate until no new pairs are formed, indicating stabilization. In the given context, \(R^3\) and \(R^4\) both stabilize, meaning the relation's powers become stable and repeat, allowing us to find the smallest integers \(m = 3\) and \(n = 4\) such that \(R^m = R^n\). Determining powers is crucial for understanding the depth of reachability and transitive properties in a relation.