What are relations? I understand that a function may be considered as a set of ordered pairs which relate the elements between two sets. I understand that a function is a subset of the cartesian product between the two sets and it can be defined by an equation like $y=x+1$ or $f(x)=x+1$, on a specified domain for $x$.
I am struggling to understand relations though. i know a function is a relation (set of ordered pairs) and a relation between to sets is nothing more than a subset of the cartesian product and that you could use $y=x+1$ or $y^2 + x^2 =1$ to define a relation.
What i dont get is why "equals" is referred to as a relation and "less than". Is the equals relation the set of ordered pairs $(x,y)$ defined by $y=x$, on some domain for x? and similarly for $y<x$?

A function is a "special" relation where there are no two pairs $(x,y_1)$ and $(x,y_2)$.
Example : the relation "father-to-child" is not a function, because a father may have more than one children.
The relation "child-to-father" is a function because a child has exactly one father. 

"Less than" is a relation but not a function : $1 < 2$ and $1 < 3$ and ...
