Example of a relation on $X$? 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 $X$.
 A: 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 :

*

*You-Feng Lin & Shwu Yeng T.Lin, Set Theory With Applications (2nd revised ed, 1981), page 65 :


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 $Dom(\mathcal R)=X$.
A: 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! 
