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Back to Relations Head

Posted 30 August 2009

OPERATIONS
ON RELATIONS

Since relations
from *A* to *B *are subsets of ,
you can form the union or intersection of any
two relations.

On the set of real numbers:

¨ The union of “ = “ and " <" is (of course!) “ ” .

¨ The intersection of “ ” and “ ” is “=”.

¨ The
union of “<” and “>”
is “not equal to”.

¨ The
intersection of “<” and
“>” is the empty relation.

¨ For all and , if and only if or (or both).

¨ For all and , if and only if both and .

The **opposite** of a relation
from *A *to
*B *is a relation called from *B *to
*A, * defined by . In other words, you get from by turning around all the pairs in .

¨ On the opposite of " >" is " <" and the opposite of " " is " ".

¨
For any set *A*,
the opposite of the equals relation on *A* is the same equals relation

¨ More generally, the opposite of any symmetric relation is the same relation.

Let * be a relation from **A *to *B* and* *a relation from *B* to *C*. The **composite
**is a relation from *A *to *C* defined this
way: if and only if there is an element satisfying and .

Computer scientists may write for

Let *A *= {1 *,*2*,
*3*, *4 *,*5},

*B *= {3 *,*5*,
*7*, *9},

*C *= {1 *,*2
*,*3*, *4, 5,* *6},

*α *= {(1, 3), (1, 5), (2, 7), (3, 5), (3, 9), (5, 7)} and

* *= {(3,1), (3,2), (3,3), (7,4), (9,4), (9,5), (9,6)}.

Then

= {(1,1), (1,2), (1,3), (2,4), (3,4), (3,5), (3,6), (5,4)}.

So
*m * *n *if
and only if in the picture there is a sequence of two arrows, the first marked *α
*and the second marked ,
from *m *to *n*.

**Theorem: **If * is a relation from **A *to *B*, * *is a relation from *B* to *C* and * *is a relation from *C *to *D*, then .

It is worthwhile to draw a picture of an example like the one to the right to understand this claim.