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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?

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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?

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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".

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