Not to be confused with
Dependency relation , which is a binary relation that is symmetric and reflexive.
In mathematics , a dependence relation is a binary relation which generalizes the relation of linear dependence .
Let
X
{\displaystyle X}
set . A (binary) relation
◃
{\displaystyle \triangleleft }
a
{\displaystyle a}
X
{\displaystyle X}
subset
S
{\displaystyle S}
X
{\displaystyle X}
dependence relation , written
a
◃
S
{\displaystyle a\triangleleft S}
if
a
∈
S
{\displaystyle a\in S}
a
◃
S
{\displaystyle a\triangleleft S}
if
a
◃
S
{\displaystyle a\triangleleft S}
finite subset
S
0
{\displaystyle S_{0}}
S
{\displaystyle S}
a
◃
S
0
{\displaystyle a\triangleleft S_{0}}
if
T
{\displaystyle T}
X
{\displaystyle X}
b
∈
S
{\displaystyle b\in S}
b
◃
T
{\displaystyle b\triangleleft T}
a
◃
S
{\displaystyle a\triangleleft S}
a
◃
T
{\displaystyle a\triangleleft T}
if
a
◃
S
{\displaystyle a\triangleleft S}
a
⋪
S
−
{
b
}
{\displaystyle a\ntriangleleft S-\lbrace b\rbrace }
b
∈
S
{\displaystyle b\in S}
b
◃
(
S
−
{
b
}
)
∪
{
a
}
{\displaystyle b\triangleleft (S-\lbrace b\rbrace )\cup \lbrace a\rbrace }
Given a dependence relation
◃
{\displaystyle \triangleleft }
X
{\displaystyle X}
S
{\displaystyle S}
X
{\displaystyle X}
independent if
a
⋪
S
−
{
a
}
{\displaystyle a\ntriangleleft S-\lbrace a\rbrace }
a
∈
S
.
{\displaystyle a\in S.}
S
⊆
T
{\displaystyle S\subseteq T}
S
{\displaystyle S}
span
T
{\displaystyle T}
t
◃
S
{\displaystyle t\triangleleft S}
t
∈
T
.
{\displaystyle t\in T.}
S
{\displaystyle S}
basis of
X
{\displaystyle X}
S
{\displaystyle S}
independent and
S
{\displaystyle S}
spans
X
.
{\displaystyle X.}
Remark. If
X
{\displaystyle X}
◃
{\displaystyle \triangleleft }
X
{\displaystyle X}
◃
.
{\displaystyle \triangleleft .}
X
{\displaystyle X}
cardinality .
Examples
edit
Let
V
{\displaystyle V}
vector space over a field
F
.
{\displaystyle F.}
◃
{\displaystyle \triangleleft }
υ
◃
S
{\displaystyle \upsilon \triangleleft S}
υ
{\displaystyle \upsilon }
subspace spanned by
S
{\displaystyle S}
equivalent to the definition of linear dependence .
Let
K
{\displaystyle K}
field extension of
F
.
{\displaystyle F.}
◃
{\displaystyle \triangleleft }
α
◃
S
{\displaystyle \alpha \triangleleft S}
α
{\displaystyle \alpha }
algebraic over
F
(
S
)
.
{\displaystyle F(S).}
◃
{\displaystyle \triangleleft }
algebraic dependence . See also
edit
