Symmetric and Transitive Relations, Equivalence Relations, and Their Relationship

This article delves into the fascinating world of symmetric and transitive relations, particularly focusing on their relationship with equivalence relations. We'll explore specific scenarios and their implications, ultimately concluding with the intriguing question of whether the numbers of certain types of relations can be the same for all finite sets.

Introduction to Relations

In the realm of discrete mathematics, relations on a set play a crucial role. These relations can exhibit various properties, including symmetry and transitivity. But how do these properties interact with the concept of equivalence relations?

Equivalence Relations on a Set

Equivalence relations on a set are a special kind of relation that satisfies three conditions:

Reflexivity: Every element is related to itself. Symmetry: If one element is related to another, then the other is also related back to the first. Transitivity: If one element is related to a second, and the second to a third, then the first is related to the third.

The number of equivalence relations on a set with cardinality n is given by the Bell number Bn. Specifically, we have m B4 15.

Constructing Symmetric and Transitive Relations

Let's consider the set A {-1012}. We are tasked with finding the number of relations in A that are both symmetric and transitive, denoted as k. The goal then is to determine the number of equivalence relations, denoted as m, and whether there exists any finite set where k and m are always the same.

Define a relation R ? A x A as symmetric and transitive on A. To extend this relation, consider the set A' A ∪ {3}, which now contains 5 elements. Construct a relation R' ? R that extends R by including all irreflexive elements from A (i.e., elements not in R) into a new cell containing the element 3. By this construction, R' remains symmetric and transitive.

The element 3 serves as a marker to gather and identify isolative irreflexive elements. This correspondence implies that the number of symmetric and transitive relations on A is exactly the same as the number of equivalence relations on A', which is k B5 52.

Generalizing the Results

To generalize, let A be a finite set with cardinality n. Then, the number of symmetric and transitive relations on A is k Bn1, and the number of equivalence relations is m Bn. However, the only n for which Bn1 Bn is n 0.

This unique value, n 0, aligns with the fact that the empty relation on the empty set is both symmetric, transitive, and an equivalence relation. For the empty set, the only relation is the empty relation itself, which is also the only equivalence relation, counted by B0 1.

Conclusion

The relationship between symmetric, transitive, and equivalence relations on finite sets is rich and nuanced. While the numbers of such relations can vary widely for different finite sets, the special case of the empty set provides a unique scenario where the counts are identical. This exploration sheds light on the intricate nature of relations and their properties.