word choice - Mathematical distinction in "Relation between edges" VS "Relationship between edges"?

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

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?

Is there a word/phrase for "unperformant"?

As a software engineer, I need to sometimes describe a piece of code as something that lacks performance or was not written with performance in mind. Example: This kind of coding style leads to unmaintainable and unperformant code. Based on my Google searches, this isn't a real word. What is the correct way to describe this? EDIT My usage of "performance" here is in regard to speed and efficiency. For example, the better the performance of code the faster the application runs. My question and example target the negative definition, which is in reference to preventing inefficient coding practices. Answer This kind of coding style leads to unmaintainable and unperformant code. In my opinion, reads more easily as: This coding style leads to unmaintainable and poorly performing code. The key to well-written documentation and reports lies in ease of understanding. Adding poorly understood words such as performant decreases that ease. In addressing the use of such a poorly ...

nouns - what do you call the person who passes the trophies to the prize giver/presenter?

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Is 'efficate' a word in English?

I routinely hear the word "efficate" being used. For example, "The most powerful way to efficate a change in the system is to participate." I do not find entries for this word in common English dictionaries, but I do not have an unabridged dictionary. I have checked the OED (I'm not sure if it is considered unabridged), and it has no entry for "efficate". It does have an entry for "efficiate", which is used in the same way. Wordnik has an entry for "efficate" with over 1800 hits, thus providing some evidence for the frequency of use. I personally like the word and find the meaning very clear and obvious when others use it. If it's not currently an "officially documented" word, perhaps its continued use will result in it being better documented.

Notes

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