1. Chapter 4 (Part 2): Mathematical Reasoning, Induction & Recursion
CSE 504 Discrete Structures & Foundations of Computer Science
Dr. Djamel Bouchaffra
Chapter 4 (Part 2): Mathematical Reasoning, Induction & Recursion
Recursive Definitions (3.4)
Ch. 3 (Part 2): Sections 3.4

2. Introduction
Define an object in terms of itself: this process is called recursion
CS 210 Discrete Structures, Chapter 4 (Part 2)

3. 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)
CS 210 Discrete Structures, Chapter 4 (Part 2)

4. Recursively Defined Functions
We use 2 steps to define a function recursively:
Basis step: specify the the value of the function at 0
Recursive step: give a rule for determining its value at an integer from its value at smaller integers
Such a definition is called a recursive or induction definition
CS 210 Discrete Structures, Chapter 4 (Part 2)

5. 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 * = 9
f(2) = 2f(1) + 3 = 2 * = 21
f(3) = 2f(2) + 3 = 2 * = 45
f(4) = 2f(3) + 3 = 2 * = 93
CS 210 Discrete Structures, Chapter 4 (Part 2)

6. 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)
CS 210 Discrete Structures, Chapter 4 (Part 2)

7. Recursively Defined Functions (cont.)
Example: Give a recursive definition of
Solution:
Basis step:
Recursive step:
CS 210 Discrete Structures, Chapter 4 (Part 2)

8. 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, …
CS 210 Discrete Structures, Chapter 4 (Part 2)

9. 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
f3 = f2 + f1 = = 2
f4 = f3 + f2 = = 3
f5 = f4 + f3 = = 5
f6 = f5 + f4 = = 8
Recursively defined sets and structures.
CS 210 Discrete Structures, Chapter 4 (Part 2)

10. 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  = 6  S (recursive step)
6  S  3  S  = 9  S (recursive step)
6  S  6  S  = 12  S (recursive step)
CS 210 Discrete Structures, Chapter 4 (Part 2)

11. Recursively Defined Functions (cont.)
Definition 2:
The set * of strings over the alphabet  can be defined by Basis step:   * (where  is the empty string containing no symbols) Recursive step: if w  * and x  , then wx  *.
Example: if  = {0, 1} Basis step:   *
Recursive step: 0, 1 (0  *, 1  *) Recursive step:  *
CS 210 Discrete Structures, Chapter 4 (Part 2)

12. Recursively Defined Functions (cont.)
Definition 3
Two strings can be combined via the operation of concatenation. Let  be a set of symbols and * the set of strings formed from symbols in . We can define the concatenation of two strings, denoted by  recursively as follows: Basis step: if w  *, then w   = w, where  is the empty string.
Recursive step: if w1  *, w2  * and x  , then
w1  (w2x) = (w1  w2) x.
Example: w1= abra w2 = cadabra
w1w2 = abracadabra

13. 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.
CS 210 Discrete Structures, Chapter 4 (Part 2)

14. Building up rooted trees
Basis step
Step 1
Step 2 … …