In general, let us denote the identity function for a set C by i C . That is, define i C : C → C to be the function given by the rule i C ( x ) = x for all x ∈ C . Given that f : A → B , we say that a function g : B → A is a left inverse for f if g ∘ f = i A ; and we say that h : B → A right inverse for f if f ∘ h = i B . Show that if f has both a left inverse g and a right inverse h , then f is bijective and g = h = f − 1 .
In general, let us denote the identity function for a set C by i C . That is, define i C : C → C to be the function given by the rule i C ( x ) = x for all x ∈ C . Given that f : A → B , we say that a function g : B → A is a left inverse for f if g ∘ f = i A ; and we say that h : B → A right inverse for f if f ∘ h = i B . Show that if f has both a left inverse g and a right inverse h , then f is bijective and g = h = f − 1 .
Solution Summary: The author explains that if f has both a left inverse and h, then it is bijective.
In general, let us denote the identity function for a set
C
by
i
C
. That is, define
i
C
:
C
→
C
to be the function given by the rule
i
C
(
x
)
=
x
for all
x
∈
C
. Given that
f
:
A
→
B
, we say that a function
g
:
B
→
A
is a left inverse for
f
if
g
∘
f
=
i
A
; and we say that
h
:
B
→
A
right inversefor
f
if
f
∘
h
=
i
B
.
Show that if
f
has both a left inverse
g
and a right inverse
h
, then
f
is bijective and
g
=
h
=
f
−
1
.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.