# Dictionary Definition

independency n : freedom from control or
influence of another or others [syn: independence]

# Extensive Definition

In mathematics and computer
science, a dependency relation is a binary
relation that is finite, symmetric,
and reflexive.
That is, it is a finite set of ordered pairs D, such that

- If (a,b)\in D then (b,a) \in D (symmetric)
- If a is an element of the set on which the relation is defined, then (a,a) \in D (reflexive)

In general, dependency relations are not transitive;
thus, they generalize the notion of an equivalence
relation by discarding transitivity.

Let \Sigma denote the
alphabet of all the letters of D. Then the independency induced
by D is the binary relation I

- I = \Sigma \times \Sigma - D

That is, the independency is the set of all
ordered pairs that are not in D. Clearly, the independency is
symmetric and irreflexive.

The pairs (\Sigma, D) and (\Sigma, I), or the
triple (\Sigma, D, I) (with I induced by D) are sometimes called
the concurrent alphabet or the reliance alphabet.

The pairs of letters in an independency relation
induce an equivalence relation on the free monoid
of all possible strings of finite length. The elements of the
equivalence
classes induced by the independency are called traces, and
are studied in trace
theory.

## Examples

Consider the alphabet \Sigma=\. A possible dependency relation is- \begin D

The corresponding independency is

- I_D=\

Therefore, the letters b,c commute, or are
independent of one-another.