Inverse relation
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In
mathematics, the
inverse relation of a
binary relation is the relation that occurs when you switch the order of the elements in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms, if
L &su(;eq X ⋅ Y
is a relation from
X to
Y then
L-1
is the relation defined so that
yL-1x
if and only if
xLy
(Halmos 1975, p. 40). In another way,
L-1 = (y x) ∈ X ⋅ Y mid (x y) ∈ L
The notation comes by analogy with that for an
inverse function. Though many functions do not have an inverse; every relation does.The
inverse relation is also called the
converse relation or
transpose relation (in view of its similarity with the
transpose of a matrix: these are the most familiar examples of
dagger categories), and may be written as
LC,
LT,
L~ or
breveL
.Note that, despite the notation, the converse relation is
not an inverse in the sense of
composition of relations:
L ο L-1 ≠q mathrmid
in general.
Properties
A relation equal to its inverse is a
symmetric relation (in the language of
dagger categories, it is
self-adjoint).If a relation is
reflexive,
irreflexive,
symmetric,
antisymmetric,
asymmetric,
transitive,
total, {{ml|Binary_relation|Relations_over_a_set|trichotomous}}, a
partial order,
total order,
strict weak order, {{ml|Strict_weak_order|Total_preorders|total preorder}} (weak order), or an
equivalence relation, its inverse is too.However, if a relation is {{ml|Binary_relation|Relations_over_a_set|extendable}}, this need not be the case for the inverse.The operation of taking a relation to its inverse gives the
category of relations Rel the structure of a
dagger category.The the set of all
binary relations
B(X) on a set X is a
semigroup with involution with the involution being the mapping of a relation to its inverse relation.
Examples
For usual (maybe strict or partial)
order relations, the converse is the naively expected "opposite" order, e.g.
(≤)-1= ≥ ~ (<)-1= >
, etc. (Parentheses would not be needed here but have been added for clarity.)
Inverse relation of a function
A function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inverse function.The inverse relation of a
function f : X → Y
is the relation
f-1 : Y → X
defined by
gra(h f-1 = (y x) mid y = f(x)
.This is not necessarily a function: One necessary condition is that
f be
injective, since else
f-1
is
multi-valued. This condition is sufficient for
f-1
being a
partial function, and it is clear that
f-1
then is a (total) function
if and only if f is
surjective.In that case, i.e. if
f is
bijective,
f-1
may be called the
inverse function of
f.
See also
References
- {{Citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | title=Naive Set Theory | isbn=978-0-387-90092-6 | year=1974}}
Inversa rilatoمعادله معکوسه
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