Mathematical distinction in "Relation between edges" VS "Relationship between edges"?

Question

This question on the distinctions between relation and relationship is general and not about mathematics. So with respect to mathematics such as graph theory: I am confused when people use the terms relation and relationships interchangeably. For example, why do some authors use relationship in the context of Euler's formula relating, faces and vertices (eg) while other authors use the term relation? What is the difference between relation and relationship in Mathematics such as graph theory? Which alternative should I choose to use more? In which context?

Answer 1

A relation between two sets is a formally defined entity. Specifically, it is a set of ordered pairs of the form (x,y) where x is in the first set and y is in the second set. As a simple example, I can define a relation between the set of all employees of a company and the set of all departments at that company by pairing employees with the department(s) they work within. Such a relation would be very important to the design of our company database, among other things.

On the other hand, the word relationship is typically used very informally with no set-in-stone definition (at least none I know of). So I could say there is a relationship between employees and departments because both are essential ingredients in the success of our company. This statement need not have a specific mathematical interpretation.

Going to graph theory specifically, a graph is a relation between at set and itself, where to put the ordered pair (x,y) into the graph means that element x is somehow connected to element y. We change our terminolgy, and call x and y nodes or vertices of the graph, and the pair (x,y) is an edge. We also illustrate graph this by drawing x and y as dots, and connect them with a line segment.

I can formally say that the nodes are parts of a relation. I can also say informally say that nodes and edges have a relationship in the formation of the graph and its illustration.

Yes, this is confusing. But have you ever seen how many different meanings the word "normal" has in various branches of mathematics?

Answer 2

I believe this is in mathematics just as in the more general language question you point to. So, you could say "there exists a relation between these vertices" (abstract fact), and specify "the relationship between vertex A and B is so and so".

Answer 3

Relations are defined in Mathematics as a subset of the Cartesian Product between sets. Relationships are defined in Linguistics as a semantic correspondence between words.

Mathematics and Linguistics cover the same pure concepts, both mention Sets which we attribute to mathematics probably because the concepts are analytical. Mathematics is a TYPE of language. There is semantics in mathematics by design, allowing us to analyze objects. In that, we are able to define different languages for mathematics, as it will be important for expanding on your question.

1. Introduction with Paradoxes:

In Category Theory, a morphism maps between a source and target element. A category is a class containing morphisms (also called arrows in the context of categories). A category satisfies these properties:

• associativity
• identity
• composition

Categories with the property of “invertibility” specify a Groupoid. Non-Categories without the properties of “associativy” nor “identity” generalize a Magmoid (magmoid is a homonym), which is also called a “Discrete Category”.

Here, specification and generalization are important.

Categories contain elements; however, categories are not Sets. This is because categories can be ”large”. “Class” is a common term we could use… Instead categories and discrete categories are referred to as ”Structures”, because the elements are being related.

“Structure” is very abstract.

A generalization of a category can be represented by a Quiver (a directed multi-graph). Graphs are “concepts with attitude”, suggesting graphs should be abstract, and convey an idea.

Therein, morphisms can be generalized as Graph Edges. (Not important here) Morphisms are directed and composable, which makes them distinct.

Main Idea (Issues/ Paradoxes):

Now we have enough information to recognize some key issues and paradoxes…

• Graphs can be referred to as Relations, but if [relations were morphisms] and [morphisms were edges], then there is a circular paradox.
• Graphs and “HyperGraphs” can be referred to as Sets, but if [graphs were sets] and [categories were graphs] and [categories were not sets], then there is a logical paradox.
• Edges can be referred to as Relations, but if [Morphisms were Edges] and [Edges were Relations] then morphisms would have to be relations…

How should we interpret these concepts then?

2. Semantics and Interpretation:

“Morphisms are not relations…”

But morphisms are mappings, and mappings are binary relations. Buttt, morphisms also come from structures, which are “concepts with attitude” — which is why I mentioned it earlier…

The properties of morphisms can be defined using Logic, without the need to define relations. This also aids in the defining of relations and other properties.

There is such a thing as a correspondence which is the generalization of a binary relation (relations have natural “arity”). This is an example of relation-like behavior before relations are even defined!

Other Examples (Pre-Relation behavior):

• Morphisms in general
• Edges/ HyperEdges in general, but the behavior is not unusual, as it can be described by Classes.
• Arrangements (combinations and permutations)
• Partitions (see MultiPartite Graphs for more information)
• Logical Expressions

Logical Expressions include topics surrounding Quantifiers, Predicates, etc…

3. Language:

Relating this discussion back to language, Relationships are not defined very well in the context of these concepts…

Everything I have mentioned is not about mathematics. The pure concepts that govern mathematics and linguistics are the same, but sometimes they have different names as we scientifically apply them across these different disciplines.

In language, Relationships create a correspondence between words. Semantically, relationships can be seen as “concepts with attitude” that capture the same behavior in Graph theory. We don’t know why edges are able to ‘relate’ the elements of graphs, if they are not technically relations. They are relationships.

In Summary: In general, Graph edges are not relations, but relationships. This is because Graphs are “concepts with attitude”. Also edges of a Graph can be generalized by relationships in linguistics. Since relations map between sets, we would require a more specific definition of a Graph to consider the edges as relations.

Many of these concepts have alternative terminology and definitions.