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I can understand "relation $R$ in $X$" through the following example in the book, but I haven't got a clue of what "relation on $X$" looks like. Can you give an example of of a relation on $X$?

"Often $A$ and $B$ are the same set, say $X$. In that case, we shall say that $R$ is a relation in $X$ instead of "from $X$ to $X$. For example, in a community $X$, to say that $a$ (for Albert) is the husband of $b$ (for Bonita), is to consider Albert and Bonita as an (ordered) pair $(a, b)$ in the relation $H$ (of being the husband of...)"

Source: Set Theory: An Intutive Approach by Shwu-Yeng T. Lin, ‎You-Feng Lin, p.137

I understand from the above explanation that $a$ relation in $X$ means $R=\{(a, b)|(a, b) \in X \times X\}$

"When the domain of a relation $R$ is obviously $X$ itself, most mathematicians prefer to say "relation on $X$" instead of "relation $R$ in $X$""

Source: Set Theory: An Intutive Approach by Shwu-Yeng T. Lin, ‎You-Feng Lin, p.143

I understand from the above explantion that a relation in $X$, i.e. $R=\{(a, b)|(a, b) \in X \times X\}$, is called a relation on $X$ when Dom($R$)$=X$

But I can't find examples of the relation on $X$.

JKnecht
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buzzee
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  • The identity map of the set $X$ is a relation on $X$. – Kavim Dec 29 '15 at 19:33
  • For $X$ equal to the set of real numbers, the following are relations on $X$ in the sense your author has defined: equality, less than, greater than, at most 18 more than, the cube root of. Personally, I think the authors are setting themselves and others up for trouble with this "in" and "on" distinction. Doing this adds a lot more to proof-reading work, and it's very easy to use/type "on" when "in" should be used (i.e. the notation is not very forgiving for the kinds of errors likely to be made). – Dave L. Renfro Dec 29 '15 at 19:38

2 Answers2

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See Binary relation :

In mathematics, a binary relation on a set $A$ is a collection of ordered pairs of elements of $A$. In other words, it is a subset of the Cartesian product $A \times A$.

As you can see, the terminology is not so "stable".

I'm not able to locate your source, but I've found :

Definition 2. A relation $\mathcal R$ from $A$ to $B$ is a subset of the Cartesian product $A\times B$. It is customary to write a $a \mathcal R b$ for $(a, b) \in \mathcal R$. The symbol $a \mathcal R b$ is read "$a$ is $\mathcal R$-related to $b$."

Often $A$ and $B$ are the same set, say $X$. In that case, we shall say that $\mathcal R$ is a relation in $X$ instead of "from $X$ to $X$"

According to this definition, an example of a binary relation "in $X$" is the relation "$x$ is sibling of $y$" on the set $X$ of humans : not all humans are siblings.

If $X = \{ John, Mary, Tom \}$ and John and Mary are siblings, but Tom is an only son, then :

$\mathcal R = \{ (John, Mary),(Mary, John) \}$

is a relation in $X$.

I'vo not found an explicit definition of relation "on $X$", but page 66 has the following example :

The equals relation $=$, on the set $\mathbb R$ of real numbers is clearly an equivalence relation.

We have that $=$ is defined for all $x \in \mathbb R$.

Thus, it is assumed that a relation $\mathcal R$ from $X$ to $X$ is "on $X$" when $Dom(\mathcal R)=X$.

Community
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Mauro ALLEGRANZA
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    The domain of $R$ is not $X$ so in the line of question $R$ should be called a relation in $X$. – drhab Dec 29 '15 at 18:56
  • The domain of R is {John, Mary}. Hence Dom(R)≠X – buzzee Dec 29 '15 at 19:23
  • Oh I found another example, but it seems that a relation in X and on X is not distinguished in Discrete Mathematics: "A relation on a set A is a relation from A to A" Source: Discrete Mathematics with Applications by Susanna Epp, p.446 – buzzee Dec 29 '15 at 19:53
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A relation $R$ from set $A$ to set $B$ is a subset of $A\times B$. When $A=B$ we say $R$ is a relation on $A$, which means $R\subseteq A \times A$. For example you can set $A$ to be members of a family, and define $R$ to be $(x,y) \in A\times A$ such that $x$ and $y$ are siblings.

If you are looking for an example of a left total relation you can pick any transformation!

Kavim
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  • I would also call $R$ a relation on $A$, but the source mentioned in the question speaks of a relation in $A$. – drhab Dec 29 '15 at 19:00
  • But members of a family include parents. So Dom(R)≠A. – buzzee Dec 29 '15 at 19:45
  • @buzzee define $R'$ to be the whole $A\times A$. i.e. $(a,b)\in R'$ if $b$ is in the same family as $a$. then $R'$ is on $A$. – Kavim Dec 29 '15 at 20:02