May 2024
Intermediate to advanced
332 pages
15h 45m
English
Content preview from Discrete Mathematics
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270
DISCRETE
MATHEMATICS'-------------------------
Corollary A function is a bijection
iff
it has a right inverse and a left inverse.
If
a
functionfhas
both a right
inverse and a left inverse, then they are equal.
I Proof I That a function is a bijection
iff
it
has a right inverse and a left inverse follows immediately from the
definition
of
bijection and the two previous Lemmas.
Now suppose
g : B
---7
A is a left inverse
off
and h : B
---7
A is a right inverse. Then
g
(b)=
g (id
8
(b))=
g
((j
0 h)
(b))=
g
(j(h
(b)))=
(g 0
j)
(h
(b))=
h (b), SO g
=h.
For a bijection
f:
A
---7
B,
we
conclude from the Corollary that
all
right inverses and left ...
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