Transitive relation.

Question

I have read transitive relation, if $a\mathrel R b$ and $b\mathrel Rc$ then $a\mathrel Rc$. But suppose we have a set of first cousins.

$$\mathrel R: \{(a,b),(a,c)\}$$

Where $a$, $b$ and $c$ are first cousins. Can anyone please help if the set $\mathrel R$ transitive?

enter image description here

Being first cousins is usually a symmetric relation as well. So if $(a, b)$ is there, then $(b, a)$ should too. — Arthur, Oct 14 '20 at 14:14
The relation you wrote, $R={(a,b),(a,c)}$ is indeed a transitive relation. This however has very little to do with an example of "a set of first cousins." The example of "first cousins" where two people are said to be "related" if they are first cousins will in general not be transitive so long as the number of people in the universal set being considered is large enough and you have at least some people who are cousins included since if $a$ is a first cousin of $b$ then you will also have $b$ is a first cousin of $a$ however $a$ is not said to be a first cousin of him/herself. — JMoravitz, Oct 14 '20 at 14:14
Now... if $a,b,c$ all happened to be first cousins of one another and your relation $R$ just happened to be defined on these three cousins but the pairs appearing in the relation have nothing to do with that... then all of this talk about cousins was completely meaningless and should never have been mentioned. — JMoravitz, Oct 14 '20 at 14:18
I have attached an image to clarify the relationship between a,b and c — Masoom Badi, Oct 14 '20 at 14:32
With elements $a,b,c$ as described in the image above and the relation defined as one person is "related" to another iff they are first cousins, then your relation should have been $R={(a,b),(a,c)\color{red}{,(b,a),(c,a)}}$ which is not a transitive relationship since for instance $(a,b)$ and $(b,a)$ are both pairs in the relation however $(a,a)$ is not a pair in the relation. Similarly $(b,a)$ and $(a,c)$ are both pairs in the relation however $(b,c)$ is not. — JMoravitz, Oct 14 '20 at 14:41
Recall... although you might see transitivity often defined using particular letters for variables, those are merely placeholders. Further, even though it is often written with three distinct letters, those three variables do not need to represent distinct elements, you could have had repeated values. In words, a relation is transitive iff for every occurrence of one thing being related to another thing and that second thing being related to a third (not necessarily different) thing that you must also have the first thing being related to the third thing. — JMoravitz, Oct 14 '20 at 14:45
Thank you for your explanation. It helped me to clarify this. — Masoom Badi, Oct 14 '20 at 15:25

score 1 · Accepted Answer · answered Oct 14 '20 at 14:13

1

Not transitive, b and c could be siblings.

answered Oct 14 '20 at 14:13

DreiCleaner

1,497

1

Or not related at all, if $b$ has cousins on both their mother's and father's side. – Arthur Oct 14 '20 at 14:14
@DreiCleaner, I have attached an image, so is this still not transitive? – Masoom Badi Oct 14 '20 at 14:40
Right, it is still not transitive. That image shows exactly what I said in my answer. $b$ and $c$ are not cousins because they are siblings. – DreiCleaner Oct 14 '20 at 15:13
If instead child $b$ was a child of daughter 1, then you would have $aRb$, $aRc$, and $bRc$ and it would be a transitive relation. Each pair would be first cousins. – DreiCleaner Oct 14 '20 at 15:14
Thanks for clarifying. – Masoom Badi Oct 14 '20 at 15:24

score 1 · Answer 2 · answered Oct 14 '20 at 16:03

1

Well, the set is not transitive. Transitive means that $aRb\wedge bRc\Rightarrow aRc$.

The point is: If the premise is not satisfied, the whole assertion is true.

answered Oct 14 '20 at 16:03

Wuestenfux

20,964

Transitive relation.

2 Answers2