Recursive definition
Margaret M. Fleck
5 March 2010
In this lecture, we see how to define functions and sets recursively, a math
ematical technique that closely parallels recursive function calls in computer
programs.
1
Announcements
Exams were returned in lecture today. If you missed lecture, we’ll also bring
them to discussion sections next week and afterwards you can get them at
office hours.
2
Recursive definitions
You’ve all seen recursive procedures in programming languages. Recursive
function definitions in mathematics are basically similar.
A recursive definition defines an object in terms of smaller objects of the
same type. Because this process has to end at some point, we need to include
explicit definitions for the smallest objects. So a recursive definition always
has two parts:
•
Base case or cases
•
Recursive formula/step
1
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Recursive definitions are sometimes called inductive definitions or (espe
cially for numerical functions) recurrence relations.
For example, recall that the factorial function
n
! is defined by
n
! =
n
×
(
n

1)
×
...
×
2
×
1. We can define the factorial function
n
! recursively:
•
0! = 1
•
n
! =
n
·
(
n

1)!
The recursive definition gets rid of the annoyingly informal “
...
” and,
therefore, tends to be easier to manipulate in formal proofs or in computer
programs.
Notice that the base and inductive parts of these definitions aren’t ex
plicitly labelled. This is very common for recursive definitions. You’re just
expected to figure out those labels for yourself.
Here’s another recursive definition of a familiar function:
•
g
(1) = 1
•
g
(
n
) =
g
(
n

1) +
n
This is just another way of defining the summation
∑
n
i
=1
n
. This par
ticular recursive function happens to have a nice closed form:
n
(
n
+1)
2
. Some
recursivelydefined functions have a nice closed form and some don’t, and it’s
hard to tell which by casual inspection of the recursive definition.
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 Spring '08
 Erickson
 Recursion, Recursive definition

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