In mathematics, a **dependence relation** is a binary relation which generalizes the relation of linear dependence.

Let
X
be a set. A (binary) relation
◃
between an element
a
of
X
and a subset
S
of
X
is called a *dependence relation*, written
a
◃
S
, if it satisfies the following properties:

if
a
∈
S
, then
a
◃
S
;
if
a
◃
S
, then there is a finite subset
S
0
of
S
, such that
a
◃
S
0
;
if
T
is a subset of
X
such that
b
∈
S
implies
b
◃
T
, then
a
◃
S
implies
a
◃
T
;
if
a
◃
S
but
a
⧸
◃
S
−
{
b
}
for some
b
∈
S
, then
b
◃
(
S
−
{
b
}
)
∪
{
a
}
.
Given a *dependence relation*
◃
on
X
, a subset
S
of
X
is said to be *independent* if
a
⧸
◃
S
−
{
a
}
for all
a
∈
S
.
If
S
⊆
T
, then
S
is said to *span*
T
if
t
◃
S
for every
t
∈
T
.
S
is said to be a *basis* of
X
if
S
is *independent* and
S
*spans*
X
.

**Remark.** If
X
is a non-empty set with a dependence relation
◃
, then
X
always has a basis with respect to
◃
.
Furthermore, any two bases of
X
have the same cardinality.