Introduction
Bijections and Cardinality
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics  Cardinality
172
Previous Lecture
Functions
Describing functions
Injective functions
Surjective functions
Bijective functions
Discrete Mathematics  Cardinality
173
Properties of Functions
A function
f
is said to be
onetoone
,
or
injective
,
if and only
if
f(a) = f(b)
implies
a = b.
A function
f
from
A
to
B
is called
onto
,
or
surjective
,
if and
only if for every element
b
∈
B
there is an element
a
∈
A
with
f(a) = b.
A function is called a
surjection
if it is onto.
A function
f
is a
onetoone correspondence
,
or a
bijection
,
if it is both onetoone and onto.
Discrete Mathematics  Cardinality
174
Composition of Functions
Let
g
be a function from
A
to
B
and let
f
be a function from
B
to
C.
The
composition
of the functions
f
and
g,
denoted by
f
○
g,
is
the function from
A
to
C
defined by
(
f
○
g)(a) = f( g( a ))
g
f
a
g(a)
f(g(a))
g(a)
f(g(a))
(f
○
g)(a)
f
○
g
A
B
C
Discrete Mathematics  Cardinality
175
89
100
90
80
70
49
0
…
79
69
50
Composition of Functions
(cntd)
Suppose that the students first get numerical grades from
0
to
100
that are later converted into letter grade.
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 Spring '10
 AndreiBulatov
 Inverse function, Finite set, Numerical Functions

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