Functions
Margaret M. Fleck
19 February 2010
This lecture starts the material in section 2.3 of Rosen.
It discusses
functions and introduces the concepts of onetoone and onto.
1
Announcements
Another reminder of the upcoming midterm. Remember to bring your ID.
(But we do have a backup plan if one or two of you forget.)
This lecture is brought to you by the number 65535. (This is 2
16
−
1, i.e.
the largest number you can store in a 16bit unsigned integer variable.)
Another useful fact is that 1000 is approximately equal to 2
10
.
This is
helpful when determining how large a number you will get when trying to
access locations in computer memory, especially for big memory sizes.
2
Functions
We all know roughly what functions are, from high school and (if you’ve
taken it) calculus. You’ve mostly seen functions whose inputs and outputs
are numbers, defined by an algebraic formula such as
f
(
x
) = 2
x
+ 3. We’re
going to generalize and formalize this idea, so we can talk about functions
with other sorts of input and output values.
Suppose that
A
and
B
are sets, then a function
f
from
A
to
B
(shorthand:
f
:
A
→
B
) is an assignment of exactly one element of
B
(i.e. the output
value) to each element of
A
(i.e. the input value).
A
is called the
domain
of
f
and
B
is called the
codomain
.
For example, let’s define
g
:
Z
→
Z
by the formula
g
(
x
) = 2
x
.
The
domain and codomain of this function are both
Z
.
1
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Notice that the domain and codomain are part of the definition of the
function, just like the input/output type declarations for a function in a
programming language. Suppose we define
h
:
N
→
N
such that
h
(
x
) = 2
x
.
This is a
different function from
g
because the declared domain and co
domain are different.
Two functions are equal if they have the same domain, the same co
domain, and assign the same output value to each input value.
The inputs and outputs to functions don’t have to be numbers and a
function doesn’t have to be defined by an algebraic formula. It’s sufficient
to describe a clear, explicit procedure for finding the output value, given the
input value. For example, we can define
s
:
{
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 Spring '08
 Erickson
 output value

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