CSE 321 Discrete Structures Winter 2008 Lecture 10 Number Theory: Primality.
CSE 321 Discrete Structures Winter 2008 Lecture 12 Induction.
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Transcript of CSE 321 Discrete Structures Winter 2008 Lecture 12 Induction.
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CSE 321 Discrete Structures
Winter 2008
Lecture 12
Induction
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Announcements
• Readings – Monday, Wednesday
• Induction and recursion– 4.1-4.3 (5th Edition: 3.3-3.4)
– Midterm: • Friday, February 8• In class, closed book• Estimated grading weight:
– MT 12.5%, HW 50%, Final 37.5%
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Induction Example
• Prove 3 | 22n -1 for n 0
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Induction as a rule of Inference
P(0) k (P(k) P(k+1)) n P(n)
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1 + 2 + 4 + … + 2n = 2n+1 - 1
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Harmonic Numbers
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Cute Application: Checkerboard Tiling with Trinominos
Prove that a 2k 2k checkerboard with one square removed can be tiled with:
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Strong Induction
P(0) k ((P(0) P(1) P(2) … P(k)) P(k+1)) n P(n)
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Player 1 wins n 2 Chomp!
Winning strategy: chose the lower corner square
Theorem: Player 2 loses when faced with an n 2board missing the lower corner square
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Induction Example
• A set of S integers is non-divisible if there is no pair of integers a, b in S where a divides b. If there is a pair of integers a, b in S, where a divides b, then S is divisible.
• Given a set S of n+1 positive integers, none exceeding 2n, show that S is divisible.
• What is the largest subset non-divisible subset of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }.
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If S is a set of n+1 positive integers, none exceeding 2n, then S is divisible
• Base case: n =1
• Suppose the result holds for n– If S is a set of n+1 positive integers, none
exceeding 2n, then S is divisible – Let T be a set of n+2 positive integers, none
exceeding 2n+2. Suppose T is non-divisible.
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Proof by contradiction
• Claim: 2n+1 T and 2n + 2 T
• Claim: n+1 T
• Let T* = T – {2n+1, 2n+2} {n+1}
• If T is non-divisible, T* is also non-divisible
/
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Recursive Definitions
• F(0) = 0; F(n + 1) = F(n) + 1;
• F(0) = 1; F(n + 1) = 2 F(n);
• F(0) = 1; F(n + 1) = 2F(n)
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Fibonacci Numbers
• f0 = 0; f1 = 1; fn = fn-1 + fn-2
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Bounding the Fibonacci Numbers
• Theorem: 2n/2 fn 2n for n 6
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Recursive Definitions of Sets
• Recursive definition– Basis step: 0 S– Recursive step: if x S, then x + 2 S– Exclusion rule: Every element in S follows
from basis steps and a finite number of recursive steps
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Recursive definitions of sets
Basis: 6 S; 15 S;Recursive: if x, y S, then x + y S;
Basis: [1, 1, 0] S, [0, 1, 1] S;Recursive: if [x, y, z] S, in R, then [ x, y, z] S if [x1, y1, z1], [x2, y2, z2] S then [x1 + x2, y1 + y2, z1 + z2]
Powers of 3
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Strings
• The set * of strings over the alphabet is defined– Basis: * ( is the empty string)– Recursive: if w *, x , then wx *
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Families of strings over = {a, b}
• L1
– L1
– w L1 then awb L1
• L2
– L2
– w L2 then aw L2
– w L2 then wb L2
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Function definitions
Len() = 0;Len(wx) = 1 + Len(w); for w *, x
Concat(w, ) = w for w *Concat(w1,w2x) = Concat(w1,w2)x for w1, w2 in *, x
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Well Formed Fomulae
• Basis Step– T, F, and s, where is a propositional variable
are in WFF
• Recursive Step– If E and F are in WFF then ( E), (E F), (E
F), (E F) and (E F) are in WFF
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Tree definitions
• A single vertex r is a tree with root r.
• Let t1, t2, …, tn be trees with roots r1, r2, …, rn respectively, and let r be a vertex. A new tree with root r is formed by adding edges from r to r1,…, rn.
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Extended Binary Trees
• The empty tree is a binary tree.
• Let r be a node, and T1 and T2 binary trees. A binary tree can be formed with T1 as the left subtree and T2 as the right subtree. If T1 is non-empty, there is an edge from the root of T1 to r. Similarly, if T2 is non-empty, there is an edge from the root of T2 to r.
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Full binary trees
• The vertex r is a FBT.
• If r is a vertex, T1 a FBT with root r1 and T2 a FBT with root r2 then a FBT can be formed with root r and left subtree T1 and right subtree T2 with edges r r1 and r r2.
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Simplifying notation
• (, T1, T2), tree with left subtree T1 and right subtree T2
• is the empty tree• Extended Binary Trees (EBT)
– EBT– if T1, T2 EBT, then (, T1, T2) EBT
• Full Binary Trees (FBT)– FBT– if T1, T2 FBT, then (, T1, T2) FBT
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Recursive Functions on Trees
• N(T) - number of vertices of T
• N() = 0; N() = 1
• N(, T1, T2) = 1 + N(T1) + N(T2)
• Ht(T) – height of T
• Ht() = 0; Ht() = 1
• Ht(, T1, T2) = 1 + max(Ht(T1), Ht(T2))
NOTE: Height definition differs from the textBase case H() = 0 used in text
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More tree definitions: Fully balanced binary trees
• is a FBBT.
• if T1 and T2 are FBBTs, with Ht(T1) = Ht(T2), then (, T1, T2) is a FBBT.
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And more trees: Almost balanced trees
• is a ABT.
• if T1 and T2 are ABTs with Ht(T1) -1 Ht(T2) Ht(T1)+1 then (, T1, T2) is a ABT.
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Is this Tree Almost Balanced?
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Structural Induction
• Show P holds for all basis elements of S.
• Show that if P holds for elements used to construct a new element of S, then P holds for the new element.
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Prove all elements of S are divisible by 3
• Basis: 21 S; 24 S;
• Recursive: if x, y S, then x + y S;
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Prove that WFFs have the same number of left parentheses as right parentheses
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Well Formed Fomulae
• Basis Step– T, F, and s, where is a propositional variable
are in WFF
• Recursive Step– If E and F are in WFF then ( E), (E F), (E
F), (E F) and (E F) are in WFF
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Fully Balanced Binary Tree
• If T is a FBBT, then N(T) = 2Ht(T) - 1
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Binary Trees
• If T is a binary tree, then N(T) 2Ht(T) - 1
If T = :
If T = (, T1, T2) Ht(T1) = x, Ht(T2) = yN(T1) 2x, N(T2) 2y
N(T) = N(T1) + N(T2) + 1 2x – 1 + 2y – 1 + 1 2Ht(T) -1 + 2Ht(T) – 1 – 1 2Ht(T) - 1
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Almost Balanced Binary Trees
Let = (1 + sqrt(5))/2
Prove N(T) Ht(T) – 1
Base case:
Recursive Case: T = (, T1, T2)
Let Ht(T) = k + 1
Suppose Ht(T1) Ht(T2)
Ht(T1) = k, Ht(T2) = k or k-1
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Almost Balanced Binary Trees
N(T) = N(T1) + N(T2) + 1
k – 1 + k-1 – 1 + 1
k + k-1 – 1 [ = + 1]
k+1 – 1