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preimagen

Preimagen, or preimage in English, is a concept in mathematics referring to the set of elements that map to a given subset under a function. If f: X → Y and B ⊆ Y, the preimage of B is f^{-1}(B) = { x ∈ X | f(x) ∈ B }. The term is not the inverse function unless f is bijective; in general, f^{-1} denotes the preimage operation, not an inverse map.

Basic properties include monotonicity and compatibility with set operations: if A ⊆ B ⊆ Y, then f^{-1}(A) ⊆ f^{-1}(B).

In topology, preimages underpin continuity: a function f: X → Y between topological spaces is continuous exactly

The concept applies broadly across set theory, topology, and analysis, and is fundamental for solving equations,

Preimages
preserve
unions
and
intersections:
f^{-1}(A
∪
B)
=
f^{-1}(A)
∪
f^{-1}(B)
and
f^{-1}(A
∩
B)
=
f^{-1}(A)
∩
f^{-1}(B).
Also
f^{-1}(Y
\
A)
=
X
\
f^{-1}(A).
For
a
single
point
y
∈
Y,
f^{-1}({y})
is
called
the
fiber
over
y.
when
the
preimage
of
every
open
set
in
Y
is
open
in
X.
In
measure
and
probability
theory,
if
f
is
measurable,
the
preimage
of
a
measurable
set
is
measurable,
enabling
the
definition
of
pullbacks
of
measures
and
the
expression
of
events
in
the
domain
as
f^{-1}(B).
inverse
problems,
and
in
describing
how
subsets
of
the
codomain
correspond
to
subsets
of
the
domain.