Last edited 8/30/2009 8:41:00 PM
RELATIONS:
EXAMPLES
Familiar examples
If A is any set,
the relation “ = “ is the subset 
of . This subset is called the diagonal of A, and may be written . See
also equation.
This is the subset of consisting of all pairs (r, s) of real numbers for
which r < s is true. In setbuilder notation, it is the subset . When we write “r < s” we are using
the symbol “ < “ as infix notation as described here. Note that saying “it is the subset ” I am describing the set, assuming you are
already familiar with “less than”; I am not defining it. If I claimed that was a definition, it would
be circular (defining it in terms of itself).
The “ ” relation on is the union of the “equals”
and the “less than” relation on .
The “ ” relation on the powerset
of any set S is the union of the proper inclusion
relation and the equals relation.
Extreme examples
The empty relation
If A and B are any sets, then the empty set is a
subset of and so it fits the definition of a relation
from A to B. If this relation is
called E, then for every element and ,
the statement “a E b” is false.
The total relation
If A and B are any sets, then is a subset of and so it fits the definition of a relation
from A to B. This is called the total relation
from A to B. If this relation is
called T, then for every element and ,
the statement “a T b” is true.
Many of the examples above use overloaded notation, including the equals,
inclusion, empty and total relations.
Consciousness-raising examples
The set is a subset of ,
and so it is a relation from the set to the set .
If we call this relation ,
then are all true statements, and all other statements of the form “ ”
are false. More about this relation here.
Consider a university at a fixed moment in time. Let S
be the set of currently enrolled students and C the set of classes currently offered. The relation E (for “enrolled in”) from S
to C is the subset . Then “s
E c” means that student s is
enrolled in class c. (This can be turned into a mathematical
relation by using student numbers as codes for the students and class codes for
the classes.)
Define the relation R on
the real numbers by: x R y if and
only if x y is a rational number. This is an equivalence
relation on . It is hard to imagine how you could picture
this relation!
Let be any positive real number. Then we define a real number r to be near a real number s if
and only if | r s | < . So if ,
then and 3.14 are near each other, but 3.1 and 3.2
are not near each other. Note that
nearness depends on a parameter . For a particular parameter , nearness is reflexive and symmetric,
but not transitive. (Transitivity is
discussed further here.)
Let S be the “is the sister of” relation on
the set of all people, and S’ be the
“is the sister of” relation on the set of all women. These are not the same relation. For example, S’ is symmetric but S is not.