Problem 6 - Applied Discrete Structures

Question

For the set of cities on a map, consider the relation \(x r y\) if and only if city \(x\) is connected by a road to city \(y .\) A city is considered to be connected to itself, and two cities are connected even though there are cities on the road between them. Is this an equivalence relation or a partial ordering? Explain.

Step-by-Step Solution

Answer

The relation is an equivalence relation because it is reflexive, symmetric, and transitive.

Step 1: Define Equivalence Relation

An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity. Reflexivity means for every element \(x\), \(x\) is related to itself. Symmetry requires that if \(x\) is related to \(y\), then \(y\) is related to \(x\). Transitivity means if \(x\) is related to \(y\) and \(y\) is related to \(z\), then \(x\) is related to \(z\).

Step 2: Check Reflexivity

The problem states that a city is considered to be connected to itself. Therefore, the relation \(x r x\) holds for every city \(x\). Thus, reflexivity is satisfied.

Step 3: Check Symmetry

If city \(x\) is connected to city \(y\) by a road, then city \(y\) is automatically connected to city \(x\) by the same road. Thus, \(x r y\) implies \(y r x\), satisfying the symmetry property.

Step 4: Check Transitivity

If city \(x\) is connected to city \(y\) by a road, and city \(y\) is connected to city \(z\) by a road, then city \(x\) is also connected to city \(z\) by a path through \(y\). Hence, \(x r y\) and \(y r z\) implies \(x r z\). Transitivity is satisfied.

Step 5: Define Partial Ordering

A partial order on a set must satisfy three properties: reflexivity, antisymmetry, and transitivity. Antisymmetry requires that if \(x r y\) and \(y r x\), then \(x = y\).

Step 6: Check Antisymmetry

Antisymmetry is not satisfied here because if city \(x\) is connected to city \(y\), and \(y\) is connected to \(x\), this does not imply that \(x = y\). They are simply two distinct cities connected by roads.

Step 7: Conclusion

Since the relation satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation. It does not satisfy antisymmetry and thus is not a partial ordering.

Key Concepts

Reflexivity

Reflexivity is one of the fundamental properties that define an equivalence relation. This property ensures that every element in the set is related to itself. In our scenario of cities and roads, reflexivity means that each city is connected to itself. Think of it this way: A city has a road to itself because you can start at any point in the city and return to it without needing to leave.

We can represent this idea mathematically as follows: For every city \(x\), the road from \(x\) to \(x\) always exists. This is expressed as \(x r x\). In practice, reflexivity makes sense because even if you drive around the block or loop back to a starting point within the city, you remain within its territory.

Why Reflexivity Matters

• Guarantees every element is included in its own relation.
• This is especially important in network or map-related scenarios where self-inclusion is assumed.
• Easy to verify in contexts where the element (or city) has inherent properties like connectivity within itself.

Symmetry

The symmetry property of an equivalence relation implies a mutual relationship between elements of a set. In the context of cities connected by roads, if city \(x\) can reach city \(y\) via a road, then city \(y\) must also be able to reach city \(x\) using that same road. This reciprocity of access is what defines symmetry in our city map.

We denote symmetry mathematically by stating that if \(x r y\) (meaning city \(x\) is connected to city \(y\)), then \(y r x\) must also be true. Roads are inherently two-way streets, unless otherwise specified, so this property follows naturally in the context of cities.

Why Symmetry is Important

• It highlights the bidirectional nature of relationships in systems.
• Ensures that the relation holds true from both perspectives in a network.
• Validates scenarios where mutual access or interaction is expected.

Transitivity

Transitivity is the third critical pillar of an equivalence relation, requiring that if a relationship exists between some elements and others, it should extend through intermediate elements. For cities, transitivity means if there is a road from city \(x\) to city \(y\), and another from city \(y\) to city \(z\), then there must be a path, possibly indirectly through \(y\), from city \(x\) to city \(z\).

Symbolically, this can be expressed as: If \(x r y\) and \(y r z\), then \(x r z\). This property ensures the chain of connectivity is maintained across a set, making any network or map where transit routes or connections are involved much more meaningful and comprehensive.

The Significance of Transitivity

• Ensures consistency in relationships throughout the network.
• Facilitates understanding and predicting indirect connections or routes.
• Makes it possible to infer relationships from known segments.