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CSE 504 Discrete Structures &
Foundations of Computer Science
Dr. Djamel Bouchaffra
Chapter 3 (Part 2):
Mathematical Reasoning, Induction
& Recursion
‹ Recursive Definitions & Structural
Induction (3.4)
Introduction
‹ Define an object in
terms of itself: this
process is called
recursion
Ch. 3 (Part 2): Sections 3.4
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CSE 504 Discrete Structures &
Foundations of Computer Science
Dr. Djamel Bouchaffra
Introduction (cont.)
‹We can use recursion to define sequences,
functions and sets
‹Example: the sequence of powers of 2 is given by
an = 2n for n = 0, 1, 2, … (explicitly defined)
a0 = 1; an+1 = 2an for n = 0, 1, 2, … (recursively
defined)
Recursively Defined Functions
‹We use 2 steps to define a function recursively:
X Basis step: specify the the value of the function at 0
X Recursive step: give a rule for determining its value at an
integer from its value at smaller integers
X Such a definition is called a recursive or induction
definition
Ch. 3 (Part 2): Sections 3.4
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CSE 504 Discrete Structures &
Foundations of Computer Science
Dr. Djamel Bouchaffra
Recursively Defined Functions (cont.)
‹Example: f defined recursively as:
f(0) = 3
f(n + 1) = 2 f(n) + 3
determine f(1), f(2), f(3) and f(4)
Solution: by definition:
f(1) = 2f(0) + 3 = 2 * 3 + 3 = 9
f(2) = 2f(1) + 3 = 2 * 9 + 3 = 21
f(3) = 2f(2) + 3 = 2 * 21 + 3 = 45
f(4) = 2f(3) + 3 = 2 * 45 + 3 = 93
Recursively Defined Functions (cont.)
‹Example: Give an inductive definition of F(n) = n!
Solution:
Basis step: Specify the initial value of this
function; F(0) = 1.
Recursive step: give a rule for determining
F(n + 1) from F(n)
(n + 1)! = (n + 1) * n! ⇒ F(n + 1) = (n + 1) F(n)
Ch. 3 (Part 2): Sections 3.4
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CSE 504 Discrete Structures &
Foundations of Computer Science
Dr. Djamel Bouchaffra
Recursively Defined Functions (cont.)
n
‹Example: Give a recursive definition of
∑a
k =0
k
Solution:
0
Basis step:
∑a
k =0
k
= a0
 n

Recursive step: ∑ a k =  ∑ a k  + a n + 1
k =0
 k =0 
n+1
Recursively Defined Functions (cont.)
‹Definition 1:
The Fibonacci numbers, f0, f1, f2, …, are defined by
the equations f0 = 0, f1 = 1, and
fn = fn-1 + fn-2
for n = 2, 3, 4, …
Ch. 3 (Part 2): Sections 3.4
4
CSE 504 Discrete Structures &
Foundations of Computer Science
Dr. Djamel Bouchaffra
Recursively Defined Functions (cont.)
‹Example: Determine the Fibonacci numbers
f2, f3, f4, f5 and f6
Solution:
Basis step: f0 = 0 and f1 = 1
Recursive step: fn = fn-1 + fn-2 ⇒
f2 = f1 + f0 = 1 + 0 = 1
f3 = f2 + f1 = 1 + 1 = 2
f4 = f3 + f2 = 2 + 1 = 3
f5 = f4 + f3 = 3 + 2 = 5
f6 = f5 + f4 = 5 + 3 = 8
Recursively defined sets and structures.
Recursively Defined Functions (cont.)
‹Question: How can sets be defined recursively?
Example: Consider the subset S of the set of
integers defined by:
Basis step: 3 ∈ S
Recursive steps: if x ∈ S ∧ y ∈ S ⇒ x + y ∈ S.
3 ∈ S (basis step)
3 ∈ S ∧ 3 ∈ S ⇒ 3 + 3 = 6 ∈ S (recursive step)
6 ∈ S ∧ 3 ∈ S ⇒ 6 + 3 = 9 ∈ S (recursive step)
6 ∈ S ∧ 6 ∈ S ⇒ 6 + 6 = 12 ∈ S (recursive step)
Ch. 3 (Part 2): Sections 3.4
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CSE 504 Discrete Structures &
Foundations of Computer Science
Dr. Djamel Bouchaffra
Recursively Defined Functions (cont.)
‹Definition 2:
The set Σ* of strings over the alphabet Σ can be
defined by
Basis step: λ ∈ Σ* (where λ is the empty string
containing no symbols)
Recursive step: if w ∈ Σ* and x ∈ Σ, then wx ∈ Σ*.
‹
Example: if Σ = {0, 1}
Basis step: λ ∈ Σ*
Recursive step: 0, 1 (0 ∈ Σ*, 1 ∈ Σ*)
Recursive step: 00 01 10 11 ∈ Σ*
Recursively Defined Functions (cont.)
‹Definition 3
Two strings can be combined via the operation of
concatenation. Let Σ be a set of symbols and Σ* the
set of strings formed from symbols in Σ. We can
define the concatenation of two strings, denoted
by ⋅, recursively as follows .
Basis step: if w ∈ Σ*, then w ⋅ λ = w, where λ is the
empty string.
Recursive step: if w1 ∈ Σ*, w2 ∈ Σ* and x ∈ Σ, then
w1 ⋅ (w2x) = (w1 ⋅ w2) x.
‹
Example:
Ch. 3 (Part 2): Sections 3.4
w1= abra
w2 = cadabra
w1w2 = abracadabra
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CSE 504 Discrete Structures &
Foundations of Computer Science
Dr. Djamel Bouchaffra
Recursively Defined Functions (cont.)
‹ Definition 4
The set of rooted trees, where a rooted tree consists of a set
of vertices containing a distinguished vertex called the root,
and edges connecting these vertices, can be defined
recursively by these steps:
Basis step: a single vertex r is a rooted tree.
Recursive step: Suppose that T1, T2, …, Tn are rooted trees
with roots r1, r2, …, rn, respectively. Then the graph formed
by starting with a root r, which is not in any of the rooted
trees T1, T2, …, Tn , and adding an edge from r to each of the
vertices r1, r2, …, rn, is also a rooted tree.
Building up rooted trees
Basis step
Step 1
Step 2
…
Ch. 3 (Part 2): Sections 3.4
…
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