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CSE
1400
Applied Discrete Mathematics
Functions
Department of Computer Sciences
College of Engineering
Florida Tech
Fall
2011
Functions
1
Counting Functions
2
Onto Functions
3
OnetoOne Functions
3
Cardinality Deﬁned
4
Inverse Function
4
Function Composition
4
Problems on Functions
5
Abstract
A function is a deterministic relation.
Functions
In college algebra and precalculus, you
have studied functions such as
y
=
x
y
=
x
2
y
=
log
x
y
=
e
x
y
=
cos
x
y
=
sin
x
In college algebra, the function concept is often explained by the
vertical line test: “If every vertical line crosses a graph at most once,
the graph represents a function.”
A given
value
x
in the
domain
of a function maps to
one and only one
value
y
in the
range
of the
function. Functions are denoted by writing
f
:
X
→
Y
where
X
is the
domain
and
Y
is the
codomain
of function
f
. The
range
G
of
f
is the subset of
Y
for which there is an
x
∈
X
such that
y
=
f
(
x
)
. The range of
f
also called the
image
of
X
and written
G
=
f
(
X
)
⊆
Y
To write
f
is a function from
X
to
Y
in
predicate
logic requires the
conjunction of two propositions.
(
∀
x
∈
X
)(
∃
y
∈
Y
)(
f
(
x
) =
y
)
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View Full Documentcse 1400 applied discrete mathematics
functions 2
x
=
1/2,
y
=
√
3/2
x
x
=
1/2,
y
=

√
3/2
y

1

1
2
1

1

1
2
1
2
1
x
2
+
y
2
=
1
is relation, but not a function.
Figure
1
: The vertical line
x
=
0.5
crosses the circle twice. The relation
y
=
±
√
1

x
2
between
x
and
y
is not a
function.
and
(
∀
x
∈
X
)(
∀
y
,
z
∈
Y
)((
f
(
x
) =
y
∧
f
(
x
) =
z
) =
⇒
y
=
z
)
The ﬁrst proposition states that every
element
x
in the
domain
maps
to some element
y
in the
range
: The function is
total
. The second
predicate states that if
x
maps to two different
names
y
and
z
, their
values
are, in fact, equal. There are several
things
worth knowing
about functions. For instance, it is useful to know
The number of different functions from
X
to
Y
that can be de
ﬁned.
Basic properties such as
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 Fall '11
 Shoaff
 Math, Counting

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