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ICS 253: Discrete Structures I
Induction and Recursion
King Fahd University of Petroleum & MineralsKing Fahd University of Petroleum & Minerals
Information & Computer Science DepartmentInformation & Computer Science Department
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Induction and RecursionICS 253: Discrete Structures I
Reading Assignment• K. H. Rosen,
Discrete Mathematics and Its Applications, 6th Ed., McGraw-Hill, 2006. • Chapter 4 Sections 4.1, 4.2 and 4.3
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Induction and RecursionICS 253: Discrete Structures I
Introduction
• Many mathematical statements assert that a property is true for all positive integers.• e.g. n! nn
• Many functions are defined based on certain “rules”• i.e. recursively.
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Induction and RecursionICS 253: Discrete Structures I
Section 4.1: Mathematical Induction
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Induction and RecursionICS 253: Discrete Structures I
Mathematical Induction
• Mathematical induction is an extremely important proof technique that is used extensively to prove results about a large variety of discrete objects. • Complexity of algorithms• Correctness of certain types of computer
programs• Theorems about graphs and trees, • A wide range of identities and inequalities…etc.
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Induction and RecursionICS 253: Discrete Structures I
Mathematical Induction
• Mathematical induction is used to prove propositions of the form: n P(n), where the universe of discourse is the set of positive integers
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Induction and RecursionICS 253: Discrete Structures I
First Form of Mathematical Induction
• Basis Step: P(1) is shown to be true• Inductive Step: The implication
P(n) P(n+1) is shown to be true for every positive integer n
• Question: Is this the same as circular reasoning???
• As a rule of inference:[P(1) n (P(n) P(n+1))] n P(n)
Inductive Hypothesis
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Prove that
Proof:
8
1
1
2
n
i
n ni
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Prove that the sum of the first n odd positive integers is n2
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Prove that n < 2n n
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Prove that 2n < n! for n 4
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Prove that n3 – n is divisible by 3 n Z+
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Use mathematical induction to show that
n=0,1,2,…
where
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,2
12
nH n
kH k
1...
3
1
2
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Use mathematical induction to show that if S is a finite set with n elements where n is a nonnegative integer, then S has 2n subsets.
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Induction and RecursionICS 253: Discrete Structures I
Examples• Use mathematical induction to prove the
following generalization of one of De Morgan's
laws: whenever A1, A2 , . . . , An are
subsets of a universal set U and n 2.
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1 1
n n
j jj j
A A
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Let n be a +ve integer. Show that any 2n 2n chessboard with one square removed can be tiled using L-shaped pieces, where these pieces cover three squares at a time as shown below
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Induction and RecursionICS 253: Discrete Structures I
Sketch of The Solution
Basis Step:
InductionStep:
InductiveHypothesis
Subsquare having the real hole
Subsquare with ‘artificial’hole at lower right corner
Subsquare with ‘artificial’hole at lower left corner
Subsquare with ‘artificial’ hole at upper left corner
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Induction and RecursionICS 253: Discrete Structures I
Section 4.2: Strong Induction and Well Ordering
• Strong induction is similar to the mathematical induction in both requiring a basis step.
• Strong induction differs in the inductive step, where we assume that the statement P(j) is true for all j k, and then prove that P holds for j=k+1.
• The validity of both mathematical induction and strong induction follow from the well-ordering property.• In fact, mathematical induction, strong induction,
and well-ordering are all equivalent principles.
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Induction and RecursionICS 253: Discrete Structures I
Strong Induction
• Basis Step: P(1) is shown to be true
• Inductive Step:
[P(1) P(2) … P(n)] P(n+1)
is shown to be true for every positive integer n
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Suppose we can reach the first and second rungs of an infinite ladder, and we know that if we can reach a rung, then we can reach two rungs higher. • Can we prove that we can reach every rung using
the principle of mathematical induction? • Can we prove that we can reach every rung using
strong induction?
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Show that if n is an integer greater than 1, then n can be written as the product of primes.
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Induction and RecursionICS 253: Discrete Structures I
Examples
• Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps.
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Induction and RecursionICS 253: Discrete Structures I
Well Ordering Property
• Mathematical induction follows from the following fundamental axiom
The Well Ordering Property: Every non-empty set of nonnegative integers has a least element
• The well ordering property can often be used directly in proofs.
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Induction and RecursionICS 253: Discrete Structures I
Section 4.3: Recursive Definitions and Structural Induction
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• Sometimes it is difficult to define an object explicitly. However, it may be easy to define this object in terms of itself. This process is called recursion.
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Induction and RecursionICS 253: Discrete Structures I
Recursive Definitions and Structural Induction• Recursion can be used to define sequences,
functions and sets• Sequences: can also be defined as
• Functions: Similar to sequences• After all, sequences are, themselves, functions!
• Sets
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2nna
1
1 0
2 1nn
na
a n
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Induction and RecursionICS 253: Discrete Structures I
Recursively Defined Functions
• Q3 pp 308: Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = –1, f(l) = 2 and for n = 1,2, ...a) f(n + 1) = f(n) + 3f(n – 1).
b) f(n + 1) = f(n)2 f(n – 1).
c) f(n + 1) = 3f(n)2 – 4f(n – 1)2.
d) f(n + 1) = f(n – l)/f(n).
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Induction and RecursionICS 253: Discrete Structures I
Recursively Defined Functions
• Give an inductive definition of the factorial function F(n) = n!.
• Give a recursive definition of
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0
n
kk
a
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Induction and RecursionICS 253: Discrete Structures I
Fibonacci Numbers
• Definition: The Fibonacci numbers, f0, f1, f2, . . . , are defined by the equations f0 = 0, f1 = 1, and fn = fn – l + fn – 2 for n = 2, 3, 4, . . . .
• Show that whenever n 3, fn > n – 2 , where
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1 5
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Induction and RecursionICS 253: Discrete Structures I
Recursively Defined Sets and Structures• In recursively defined sets, the basis step defines
some initial elements and the recursive step defines a rule for constructing new elements from those already in the set.
• Recursive definitions may also include an exclusion rule, which specifies that a recursively defined set contains nothing other than those elements specified in the basis step or generated by applications of the recursive step.• We will assume that it always holds.
• To prove results about recursively defined sets we use a method called structural induction.
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Induction and RecursionICS 253: Discrete Structures I
Example
• Consider the subset S of the set of integers defined by • Basis step: 3 S.• Recursive step: If x S and y S, then
x + y S.
• Show that S is the set of all positive integers that are multiples of 3.
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Induction and RecursionICS 253: Discrete Structures I
Strings Over Alphabet
• Definition: The set * of strings over the alphabet can be defined recursively by• Basis step: * (where is the empty string
containing no symbols).• Recursive step: If w* and x, then wx*.
• Example: What are the strings formed over ={0,1}?
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Induction and RecursionICS 253: Discrete Structures I
Concatenation of Strings
• Definition: 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* and w2* and x,
then w1 · (w2 x ) = (w1 . w2)x.
• Usually, w1 . w2 is denoted by w1w2
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Induction and RecursionICS 253: Discrete Structures I
Length of a String
• Give a recursive definition of l(w), the length of the string w.
• Solution:
The length of a string can be defined by• l() = 0;• l(wx) = l(w) + 1 if w * and x .
• Use structural induction to prove that l(xy) = I(x) + l(y).
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Induction and RecursionICS 253: Discrete Structures I
Well-Formed Formulae• Well-Formed Formulae for Compound Statement Forms
• We can define the set of well-formed formulae for compound statement forms involving T, F, propositional variables, and operators from the set {,,,,} as follows
• Basis step: T, F, and s, where s is a propositional variable, are well-formed formulae.
• Recursive step: If E and F are well-formed formulae, then (E), (E F), (E F), (E F), and (E F) are well-formed formulae.
• Is (p q) (q F) a well-formed formula?• Is p q a well-formed formula?• Can you similarly define well-formed formulae for operators
and operands?• Show that every well-formed formulae for compound
propositions contains an equal number of left and right parentheses.
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Induction and RecursionICS 253: Discrete Structures I
Rooted Trees
• Definition: 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
disjoint rooted trees with roots rl , 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. .
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Induction and RecursionICS 253: Discrete Structures I
Building Up Rooted Trees
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Induction and RecursionICS 253: Discrete Structures I
Extended Binary Trees
• Definition: The set of extended binary trees can be defined recursively by these steps:• Basis step: The empty set is an extended binary tree.
• Recursive step: If T1 and T2 are disjoint extended binary trees, there is an extended binary tree, denoted by Tl · T2 , consisting of a root r together with edges connecting the root to each of the roots of the left subtree Tl and the right subtree T2 when these trees are nonempty.
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Induction and RecursionICS 253: Discrete Structures I
Building Extended Binary Trees
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Induction and RecursionICS 253: Discrete Structures I
Full Binary Trees
• Definition: The set of full binary trees can be defined recursively by these steps:• Basis step: There is a full binary tree consisting
only of a single vertex r.
• Recursive step: If Tl and T2 are disjoint full binary trees, there is a full binary tree, denoted by Tl · T2 , consisting of a root r together with edges connecting the root to each of the roots of the left subtree Tl and the right subtree T2 .
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Induction and RecursionICS 253: Discrete Structures I
Examples of Full Binary Trees
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Induction and RecursionICS 253: Discrete Structures I
Full Binary Trees
• Denote the number of nodes in full binary tree T as n(T).• Can you provide a recursive definition of n(T)?
• Definition: The height h(T) of a full binary tree T recursively as follows• Basis step: The height of the full binary tree T
consisting of only a root r is h(T) = 0.
• Recursive step: If T1 and T 2 are full binary trees, then the full binary tree T = T1 · T2 has height h(T) = 1 + max(h(T 1 ), h(T2))
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Induction and RecursionICS 253: Discrete Structures I
Full Binary Trees
• Theorem: If T is a full binary tree, then
Proof (By Structural Induction):
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( ) 1( ) 2 1h Tn T
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Induction and RecursionICS 253: Discrete Structures I
Full Binary Trees
• Definition: A leaf node is a node that has no children.
• Definition: An internal node is a node that has one or more children.
• Prove that in any full binary tree, the number of leaf nodes is one more than the number of internal nodes.
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Induction and RecursionICS 253: Discrete Structures I
Generalized Induction
• Instead of using induction on Z+ or N, we can extend it to any set having the well-ordering property
• For example, consider N N with the following property (called lexicographic ordering)• (x1, y1) is less than or equal to (x2, y2) if either x1<x2,
or x1=x2 and y1<y2;
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Induction and RecursionICS 253: Discrete Structures I
Example
• Suppose that am,n is defined recursively for (m, n) N N
Show that am,n = m + n(n + 1)/2 for all (m,n)N N
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, 1,
, 1
0 if 0
1 if 0and 0
if 0m n m n
m n
m n
a a n m
a n n