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Posted 6 February 2016
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).
Terminology
Definition: A relation from a set A to a set B is a subset of A ´ B . (This is the strict definition.
See fine points.)
 Any subset of A ´ B is a relation from A to B.
 A and B can be different sets.
Examples
The relation “<”
The relation “<” on the real numbers is the subset of R ´ R 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 {(r , s ) | r < s}. 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 non-circular definition of “<” is that it is
{(r ,s) | $ t > 0 such that s = r + t }
An arbitrary relation
The arbitrary relation {(1,5),(1,6),(2,5),(2,7)} is a subset of {1,2}´ {5,6,7,8}, and so it is a relation
from the set {1,2}to the set {5,6,7,8}. I made it up to illustrate that relations can be arbitrary.
Relations on a single set
If a is a relation from a set A to A, in other words a Î A ´ A , then we say a is a relation on A.
Most of the relations studied here in abstractmath.org are relations on a single set.
How we write relations
If a Í A´ B (so a is a relation from A to B), and (a, b) Î a , then we write “ a a b ”, using infix
notation.
“ a a b ” is a statement that means that a is in the relationship a to b.
Examples
If we use g for the abritrary relation above, then 1g 5 and 2 g 7 , but it is not true that 2 g 8 . In
fact no element of A is related by g to 8. ( g is the Greek letter gamma. Don’t call it “r” or you will be
sneered at by rude math majors.)
Remarks
 The definition of relation may seem weird at first. It says in effect “a relation a is the set of pairs
(a, b) for which the statement “ a a b ” 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.
 “ a ” 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.
Two fine points
Strict and less strict
Here are three definitions of a relation. Note that they all have the same set of ordered pairs.
 g = {(1,5),(1,6),(2,5),(2,7)} from the set {1, 2} to the set {5,6,7,8}.
 g ' = {(1,5),(1,6),(2,5),(2,7)}from {1,2,3} to {5,6,7}.
 g '' = {(1,5),(1,6),(2,5),(2,7)} on the set of natural numbers N .
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 A ´ B .
 According to the less strict definition, g, g ' and g'' 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.
Relations given by rule
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.
Rules of inference for relations
Method: Let A and B be sets and let a Í A´ B .
 If you know (a, b) Î a , then you know a a b .
 If you know a a b , then you know (a, b) Î a .
Worked Exercises
Write all ordered pairs in the relation a on the set A.
1) A = {1,2,3}, a is " ¹ ".
2) A = {1,2,3,4,5}, a is "divides".
3) A = {1,2,3}, m a n if and only if n = 3.
4) A = {1,2,3}, m a n if and only if m < n + 1.
5) A = {1,2,3}, m a n if and only if m > n + 1.
Answers:
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)
Representing relations
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.
5
1
1 5
6
2
1 6
7
2 5
8
2 7
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.
y
x
This can’t be done for all relations on the reals. Try
picturing the rational difference relation, for example.
Metaphors and Images for relations
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 “ a is a subset of A ´ B ”, but you don’t normally write “ (a, b) Î a ”,
you write “ a a b ”. 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.