According to http://en.wikipedia.org/wiki/Binary_relation it is first defined as "a collection of ordered pairs of elements of A" and then as "an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) and codomain (or the set of destination), respectively, of the relation, and G is called its graph." What is the correct definition of it? Also according to http://en.wikipedia.org/wiki/Function_%28mathematics%29 a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. How a function can be a relation between sets? Shouldn't it be a relation on sets? Because defined like this, one can think of a function as an object that relates an input x to output f(x) while what it really is is an ordered triple (X, Y, G). Is there anything significant behind these definitions or it's just abuse of definitions and words?
[Math] the definition of a binary relation
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They do not seem incompatible to me since they talk about different types of 'totality'. The first definition takes 2 sets $X, Y$. While the second definition uses only one set. (It's a binary relation over one set, wikipedia speaks of endorelation)
You could transform the first definition so that it uses one set:
A relation $R \subset X\times X$ is total if it associates to every $a \in X$ at least one $b \in X$; that is
$$\forall a \in X, \exists b \in X: (a,b) \in R$$
However, this is weaker than the second definition. Wikipedia would speak of left-total.
Roughly said:
The first definition demands that from every element in the source at least one relation departs.
While the second definition demands that every element in a set has a connection with every other element in either one, or another direction (or both)
Compare following examples: In the first picture the relation is left-total from $X$ to $X$. (Every element has at least one arrow departing - C has even two arrows departing). While the (endo)relation is not total as in definition 2. In the second picture it is total as in in definition 2. It also seems to be left-total.
(When a relation is total as in definition 2, it does not have to be left-total. Can you find an example?)
Assume we have such a relation. It is symmetric so xRy implies yRx. It is antisymmetric so xRy and yRx implies x=y. But putting this together we get xRy implies x=y. Thus our relation is the identity function over some set. But the identity function is transitive vacuously. This is a contradiction.
Best Answer
You have many questions, I'll try to adress them all.
A binary relation, as you read is just some set $R$ which is a subset of the cartesian product of two sets $A$ and $B$, that is $R \subseteq A\times B$.
An example may ilustrate this:
Let $A=\{\dots,-4,-2,0,2,4,\dots \}$ (the set of even numbers), $B=\{1,3,5\}$.
Then a relation $R_1$ could be $R_1=\{(-4,1),(-4,3)(0,5)\}$
We usually denote a pair $(a,b)$ of a relation with the notation $aRb$ meaning a is related with b.
A function is a relation between two elements of two given sets condition that for each element in the domain there's one and only one image(*).
(*)That is: if $R$ is a function, $x_1\in Dom(f)$ and $y_1,y_2\in Im(f)$, $$ x_1Ry_1 \wedge x_1Ry_2 \iff y_1=y_2 $$