The Universe of Discourse
           
Thu, 17 Apr 2008

Is blood a transitive relation?
When you're first teaching high school students about the idea of a relation, you give examples of the important properties of relations. Relations can be some, none, or all of reflexive, symmetric, antisymmetric, or transitive. You start with examples that everyone is already familiar with, such as the identity relation, which is reflexive, symmetric, and transitive, and the ≤ relation, which is antisymmetric and transitive. Other good examples include familial relations: "sister-in-law of" is symmetric on the set of women, but not on the larger set of people; "ancestor of" is transitive but not symmetric.

It might seem at first glance that "is related to" is transitive, but, at least under conventional definitions, it isn't, because my wife is not related to my cousins.

(I was once invited to speak at Haverford College, and, since I have no obvious qualifications in the topic on which I was speaking, I was asked how I had come to be there. I explained that it was because my wife's mother's younger brother's daughter's husband's older brother's wife was the chair of the mathematics department. Remember, it's not what you know, it's who you know.)

I think I had sometimes tried to turn "related to" into a transitive relation by restricting it to "is related to by blood". This rules out the example above of my wife being unrelated to my cousins, because my relationship with my wife is not one of blood. I don't quite remember using "related by blood" as an example of a transitive relation, but I think I might have, because I was quite surprised when I realized that it didn't work. I spent a lot of time that morning going over my counterexample in detail, writing it up in my head, as it were. I was waiting around in Trevose for the mechanic to finish examining my car, and had nothing better to do that morning. If I had had a blog then, I would probably have posted it. But it is a good thing that I didn't, because in spite of all my thought about it, I missed something important.

The example is as follows. A and B have a child, X. (You may suppose that they marry beforehand, and divorce afterward, if your morality requires it.) Similarly, C and D have a child, Z. Then B and C have a child Y. Y is now the half-sibling of both X and Z, and so is unquestionably a blood relative of both, but X and Z are entirely unrelated. They are not even step-siblings.

Well, this is all very well, but the reason I have filed it under oops/, and the reason it's a good thing I didn't post it on my (then nonexistent) blog is that this elaborate counterexample contains a much simpler one: X is the child and hence the blood relative of both A and B, who are not in general related to each other. C, D, Y, and Z are wholly unnecessary.

I wish I had some nice conclusion to draw here, but if there's something I could learn from it I can't think would it might be.


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