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Back to Relations Head
Posted 30 August 2009
RELATIONS:
BASIC IDEAS
You are familiar with relations such as “ = “
and “ < “ that may hold between numbers.
The mathematical concept of relation includes these but is much more general and abstract than you might expect.
A
relation need not have a special symbol such as “=” or “<”.
A
relation can hold between two different kinds of things.
A
relation can be abstract and arbitrary.

The moderately devious definition I give below captures this
generality, but it
is hard to get your head around the definition when you first see it. I suggest reading it and then looking at the
examples to see how it works.
Note: This site discusses only relations between two objects (binary relations).
Definition: A relation from a set A to a set B is a subset of . (This is the strict definition. See fine points.)
¨ Any subset of is a relation from A to B.
¨ A and B can be different
sets.
The relation “<” on the real numbers is the subset of consisting of all pairs (r, s) of real numbers for which r < s. This means that the relation “<” is a mathematical object.
In setbuilder notation, the
relation “<” is the subset . If this were a definition of “<” it would be circular. In fact it is an illustration of the definition of relation and assumes you already
know what “<” means. One possible noncircular
definition of “<” is that it is
The arbitrary relation is a subset of , and so it is a relation from the set to the set . I made it up to illustrate that relations can be arbitrary.
If is a relation from a set A to A, in other words , then we say is a relation on A. Most of the relations studied here in abstractmath.org are relations on a single set.
If (so is a relation from A to B), and , then we write “ ”, using infix notation.

If we use for the abritrary relation above, then and , but it is not true that . In fact no element of A is related by to 8. ( is the Greek letter gamma. Don’t call it “r” or you will be sneered at by rude math majors.)
¨ The
definition of relation may seem weird at first.
It says in effect “a relation is the set of pairs (a, b) for which the statement “ ” is true”.
It has the advantage that you don’t have to describe in words what makes
the relationship hold, you simply have to say what pairs it is true of. This allows relations to be arbitrary. This is by far the most common way to define relations
in math, but there are other ways.
¨ “r < s” is a symbolic
assertion but “r + s” is only an expression (symbolic term) that denotes another number. It makes sense to ask whether “r < s” is true or false but it does not make sense to ask whether “r + s” is true or
false. Note that the symbol “<” by
itself is a symbolic term.
¨ “ ” is the Greek letter alpha. In abstractmath.org I often use Greek letters for relations if there is not already a symbol such as “<” or “ ” for the relation.
Here are three definitions of a relation. Note that they all have the same set of ordered pairs.
¨ from the set {1, 2} to the set {5,6,7,8}.
¨ from {1,2,3} to {5,6,7}.
¨ on the set of
natural numbers .
Are these the same relation or different
relations? Mathematical usage differs on
this point.
The less strict definition of relation: A relation is a set of ordered pairs.
The stricter definition of relation from A to B: A relation from A to B is a subset of .
¨ According to the less strict definition, are all the same relation.
¨ According
to the stricter definition they are all different.
This
distinction is closely analogous to the distinction between the two definitions of function.
Remarkably
often, it makes no difference which definition is used in a particular context
and the author of a text might not tell you what stance she is taking on this
point.
In some cases
you must assume they are different. This is true of the definition of reflexive, for example.
Now
look at the equals relations
on two different sets:
¨
On the set A = {1, 2} the equals relation is {(1, 1), (2, 2)}.
¨
On B = {2, 3, 4} it is {(2, 2), (3, 3), (4, 4)}.
The
definition of the equals relation is the same in both cases, in fact on any set: a = b if and only if
a and b are the same mathematical
object. But the two
relations given above are nevertheless different relations, by either the strict or the less strict definition,
because they are different sets of ordered pairs.
Thus “equals”
is the name of many different relations, depending on the set it is defined
on. Another example of this phenomenon
occurs with the “is
the sister of” relation. More
about this under overloaded
notation.
Method: Let A and B be sets and let .
¨
If you
know ,
then you know .
¨
If you
know ,
then you know .
Write all ordered pairs in the relation on the set A.
1) A = {1,2,3}, is " ".
2) A = {1,2,3,4,5}, is "divides".
3) A = {1,2,3}, if and only if n = 3.
4) A = {1,2,3}, if and only if m < n + 1.
5) A = {1,2,3}, if and only if m > n + 1.
1) (1,2), (1,3), (2,3), (2,1), (3,1), (3,2)
2) (1,1), (1,2), (1,3), (1,4), (1,5), (2,4)
3) (1,3), (2,3), (3,3)
4) (1,1), (1,2), (1,3), (2,2), (2,3), (3,3)
5) (3,1)
Relations between small finite sets can be represented graphically, with arrows from a to b if a is related to b, and also by a table. Here are a graph and a table representing the arbitrary relation given above.
In some cases, a relation on the real numbers can be displayed by shading the area containing the pairs for which the relation holds. The graph below shows part of the relation “<” on the reals.
This can’t be done for all relations on the reals. Try picturing the rational difference relation, for example.
If you are new to abstract math you perhaps think of a relation as some sort of natural connection between certain pairs of objects. The kinds of examples you think of are like “<” on numbers or “is the sister of” on the set of all women.
The definition of relation as a set of ordered pairs
says nothing about this, for the good reason that it is really really hard to
say what “natural connection” means. This definition is an abstraction of the intuitive idea of
relationship. A relationship is
something that holds between two objects, so you define relation to be any set of ordered pairs of objects.
This definition also suits mathematicians, who generally want
all their concepts to allow complete
arbitrariness within the
bounds of the definition.
So you get this
weird thing that “ is
a subset of ”, but you don’t normally write “ ”, you write “ ”.
Indeed most of the time your mental image of a relation does not include thinking about
it as a set (of ordered pairs or leprechauns or anything) at all. “Is the sister of” is a SET??
(Functions can
cause this difficulty too, but the ordered pairs part is not so hard to deal
with because you are used to thinking of the graphs of functions.)
So all I can
say is
Get used to it: A relation
is a set of ordered pairs.