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RELATIONS: EXAMPLES

Familiar examples

Equals

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.

Less than (for real numbers)

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).

Less than or equal

The “  ” relation on  is the union of the “equals” and the “less than” relation on .

Inclusion

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.

Remark

Many of the examples above use overloaded notation, including the equals, inclusion, empty and total relations.

Consciousness-raising examples

An arbitrary relation

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.

Students and classes

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.)

Rational differences

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!

Nearness

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.)

Sister

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.