Can Relations Have Only One Variable? Are equations with only one variable such as:
$x = x^2$
and 
$x = 5$
considered relations? Do they have domains and codomains?
 A: Short answer: no. 
Longer answer. A relation is a set of pairs of things. The set where you can find the first elements of the pair is the domain,  the second the codomain. Without the "pair" you can't make sense of that. 
Although "domain" and "codomain" make sense for any relation, they are most commonly used for relations that happen to be functions.
Thinking about "variables" in equations is not a good way to understand this material. The language of sets is better.
A: Short answer. 
An equation is not a set, it is a sentence ( an open sentence) so it cannot be, as such, a relation, for a relation is a set. 
Now, an equation can "define" a relation. 
Although a relation is a set of pairs (x,y) you do not need an equation in two variables to define a relation. 
Define in the cartesian plane the following relation 
D = { (x,y) |  x = 10 }. 
Any point ( pair (x,y) ) with x=10 belongs to this relation, whatever y may be. 
this relation is simply the vertical line that intersects perpendicularily the X axis at point ( 10,0) . 
This relation is from R to R. It is a subset of the cartesian product : R cross R. 
The domain of this relation is R ( set of real numbers) 
The codomain is {10} the singleton having 10 as unique element. 
Indeed, codomain of the relation D = the set of all y belonging to R such that for some x belonging to R , xDy is true. 
The number 10 is the only number that satisfies this condition ( for our relation D) 

Abusively, one can sometimes read things such as 
" the line : x = 4 " 
or " the line : y= -6". 
A more rigorous way of speaking would be : "the line defined by the equation : x = 4" , or " the line defined by the equation : y = - 6 ". 
A straight line ( in the cartesian plane) is a relation ( or "represents a relation") , that is, it is a set of points (x,y) all belonging to the cartesian product of R by itself, R being the set of real numbers). So it is a subset of "R cross R" ( cartesian product ...) 
Now  what is this ( informally defined)  relation  x = x² ? 
It is the set of all points (x,y) such that x= x². 
That is the set { (x,y) | x = x² } 
Note that here, no condition is imposed on the y-coordinate. It is only on the x-cordinate that a condition is imposed. 
Lets decipher this condition by solving the equation for x 


*

*If x = x² 

*Then : x² = x 

*Then : x²-x = O

*Then : x(x-1) = O 

*Then : x = 0 OR x = 1 
Let's check. 
(1) if x= 0, then x²=0=x --> OK 
(2) If x = 1 , then x²=1² = 1=x --> OK 
The set of all points (x,y) such that x= 0 is simply the Y axis of the cartesian plane. 
The set of all points (x,y) such that x = 1 is vertical line that intersects perpendicularly the X axis at point ( 1, 0). indeed, all points of this line have an X-cordinate of 1 ( whatever their Y-cordinate may be). 
The set that is defined by the equation is  : x = x² is the set of all points (x,y) such that :
x = 0 OR  x =1. 
This set is not a line, it is the set of all points that belong either to one line, or to the other. It is the UNION of these two line-sets ( so to say). 
The case : x = 5 is easier. 
Try to find which set it defines. It is the set of all points (x,y) such that ... 
This set is a line. 
Try to find which one and to draw it. 
A last question : what is the relation defined by the equation : y = O ? 
And by the equation : y = - 10? 
