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If you define a relation on the set of people, given by $R=\{x,y : x\text{ is married with } y\}$. Is this relation transitive?

I would say it depends: In the western culture: If $x$ is married with $y$ and $y$ is married with $z$ then $z$ has to be $x$ and thus $x$ is also married with $z$. In cultures where you can be married to multiple persons it is not because $z$ is not necessarily $y$.

Am I right?

AlexConfused
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    I don't think one considers people to be married to themselves in Western culture. –  Oct 30 '14 at 14:13
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    I did not ask about reflexivity. – AlexConfused Oct 30 '14 at 14:14
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    In Western culture, George H. Bush is married to Barbara Bush, and Barbara Bush is married to George H. Bush. Is George H. Bush also married to George H. Bush? – MJD Oct 30 '14 at 14:15
  • I suppose I should have quoted "$z$ has to be $x$ and thus $x$ is also married with $z$" from your question. If $z$ is $x$, then $x$ is married to $x$, ergo my comment. –  Oct 30 '14 at 14:17
  • Give me, please, one example where the marriage is not reflexive. – Karel Macek Oct 30 '14 at 14:19
  • thank you MJD, it really sucks to be stupid :( – AlexConfused Oct 30 '14 at 14:26
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    While expressed in mathematical language, I do not think this question is about mathematics. – RghtHndSd Oct 30 '14 at 15:53
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    @Alex I don't think you should describe it as stupidity. These kinds of definitions are in some ways very artificial, and it can take a lot of practice to be able to find the right examples or counterexamples quickly. In a very similar case, I once spent about twenty minutes working up an elaborate counterexample, involving two sets of half-siblings, to the claim that “$X$ is a blood relative of $Y$” is transitive, before realizing that a much simpler counterexample was the case of myself, my wife, and our daughter. – MJD Oct 30 '14 at 16:48
  • @MJD Are you saying that any relation which is symmetric and irreflexive cannot be transitive? – Max May 14 '19 at 12:58
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    @Max No, I am not saying that. See https://math.stackexchange.com/questions/1592652/example-of-a-relation-that-is-symmetric-and-transitive-but-not-reflexive – MJD May 14 '19 at 15:21
  • @MJD I don't quite understand why you mention that marriage is symmetric and ask whether it is also reflexive. (Which it isn't since it is irreflexive) In what way does the answer to this question have anything to do with whether marriage is transitive? So basically, what's the argument here, if it is one. – Max May 15 '19 at 16:14
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    @Max Assume that marriage is symmetric. If, in addition, marriage is transitive "$A$ is married to $B$" implies that "$B$ is married to $A$" (by symmetry), which further implies that "$A$ is married to $A$" (by transitivity). This contradicts irreflexivity, hence the relation is not transitive. – Xander Henderson May 15 '19 at 16:25
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    @max OP asked if a particular relation was transitive. I provided a single counterexample, showing that it was not. I said nothing about reflexivity or irreflexivity and nothing about any other properties that it might or might not have. All I did was provide the simplest possible proof that it was not transitive, in the form of a triple $(A,B,C)$ where $(A,B)$ and $(B,C)$ are in the relation but $(A,C)$ is not. – MJD May 16 '19 at 01:35
  • @Xander Henderson Exactly. The point I was trying to make was that this argument seems not to be specific to marriage, because it works for all relations which are symmetric and irreflexive. – Max May 16 '19 at 17:17

2 Answers2

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The first comment to the question says, "I don't think one considers people to be married to themselves in Western* culture". In particular, marriage is never reflexive. It is also usually symmetric. These fact are relevant because:

Lemma. Let $\sim$ be a transitive relation on a set $S$. For $a, b\in S$, if both $a\sim b$ and $b\sim a$ then $a\sim a$.

The proof is obvious.

Okay, so lets define "marriage" to be as general as possible: $A$ is married to $B$ if either $A$ or $B$ claims that this is true, or some government claims that this is true. Now, I (personally) do not claim to be married to myself, and nor does my government claim it. However, my government claims that I am married to someone, and it also claims that said someone is married to me. Hence, I am a counter-example to transitivity:

  • $\text{user1729 $\not\sim$ user1729}$
  • $\text{user1729 $\sim$ someone}$
  • $\text{someone $\sim$ user1729}$

Hence, by transitivity $\text{user1729 $\sim$ user1729}$, a contradiction.

*The OP is from Göttingen, Germany (according to their profile), so is based in the "West".

user1729
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A large part of the difficulty with this problem is that it is not well-posed. The relation "$A$ is married to $B$" is ill-defined, and depends on a great deal of non-mathematical interpretation and assumptions. For example, many societies are polygynous (one man, several women). In such a society, if a man is married to several women, are those women considered to be married to each other? The nature of the relation defined by the phrase "$A$ is married to $B$" depends on how this question is answered.

It is also possible to conceive of a society where this relation is not even symmetric—perhaps several women can be said to have married one man, but that man cannot be said to be married to all of those women. I have in mind the vows that Catholic nuns take, in which they are said to marry Jesus. Is that version of marriage symmetric? I honestly don't know the answer to this question, but it is the kind of thing which pops up.

So, in answering the question "Is marriage a transitive relation?" it is first necessary to define what "marriage" even means. For the purposes of this question, I am going to adopt the following, very narrow definition, which more-or-less conforms to current legal definition of marriage in the US:

In the United States, marriage is a legal arrangement between two individuals. When two individuals enter into a marriage, they marry each other (hence if $A$ is married to $B$, then $B$ is also married to $A$). Moreover, if $A$ is married to $B$, and $A$ is married to $C$, then $B = C$ (this is the legal requirement that a person may not be married to more than one other person). Finally, a person cannot be said to be married to themselves (the relation is irreflexive).

Under this definition, the marriage relation is not transitive. If $A$ is married to $B$, then symmetry implies that $B$ is married to $A$. If marriage were transitive, then $A$ would be married to $A$. However, we have defined marriage to be irreflexive, hence $A$ cannot be married to $A$, which implies that marriage is not transitive.

That being said, let me reiterate that this is not really a question about mathematics. It can be made into a question about mathematics by writing a reasonable definition of marriage, but the answer is not unique. For example, maybe marriage is simply illegal or doesn't exist at all (the Na of China are a purported example of the latter). If marriage is doesn't exist, then marriage is transitive by vacuity. Or maybe society is polynadrous, but marriage is intransitive because the several brother-husbands of a given woman are not considered to be married to each other. Indeed, given any possible permutation of the axioms of an equivalence relation, I can probably describe a hypothetical (or real) society in which there is a version of marriage satisfying those axioms.

Xander Henderson
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    "It is also possible to conceive of a society where this relation is not even symmetric." Those poor anime fans... they may be married to their waifu but in reality their waifu doesn't even know of their existence (or even exist themselves at all really). – JMoravitz May 17 '19 at 14:00