# What is the definition of a binary relation?

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?

• I would recommend you split this question into one about equivalence relations and one about functions. Commented May 25, 2015 at 1:29

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 $y_1,y_2\in Im(f)$, $$x_1Ry_1 \wedge x_1Ry_2 \iff y_1=y_2$$

• I don't understand. How can a function be a relation between two elements of two given sets when a relation is a set? Commented May 25, 2015 at 2:33
• Do you understand the concept of a cartesian product? To give you an example, lets say we define a function $f: \Bbb{N} \rightarrow \Bbb{Z}: f(x)=-x^2$. We can denote this function as a set of ordered pairs of the form $(x,f(x))$, in this case: $(x,-x^2)$. The set being some set $R=\{(1,-1),(2,-4),(3,-9),...\}$ describes completely this function. Commented May 25, 2015 at 2:51
• It seems I am not so good with words. Lets try this way: We define the word relation to denote a subset of the cartesian product $A \times B$. Now you say that a function is a relation between two elements of two given sets, in other words, "a set between two elements of two given sets" and i think this has no meaning. If a function is a relation (subset of $A \times B$), then you can say that an element (a,b) is in the set, where a is an element of A and b of B, and that 'a' relates to 'b'. Am I right? Commented May 25, 2015 at 3:03
• Yes, the only thing missing (and the most important to the concept of a function) is that for every $a \in A$ there exists a unique $b \in B$ such that $f(a)=b$, that is why a relation like x=y^2 is not a function. I'm sorry if I made my answer unclear, I'll try to rephrase it. Commented May 25, 2015 at 3:07