Proof Techniques

Discrete Structures (CS 173)

Madhusudan Parthasarathy, University of Illinois

Landscape with Houseand Ploughman

Van Gogh

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Proof by induction

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Prove instead its equivalent:

­ � 0 ⋀

∀� � 0���∀��� � � � ⇒ � � � ⇒ � � �

Proof by induction

Simple math example

Claim: ∑ � �������

����� for all natural integers �.

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Simple math example

Claim: ∑ � �������

����� for all natural integers �.

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Does domino n fall?

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Does domino n fall?

• Suppose dominoes 1,.. k fall. Then domino k+1 falls.

Typeequationhere.

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Does domino n fall?

• Suppose the first k dominoes fall.

Then domino k+1 falls.

fall 1 ���… fall � → fall�� � 1�• The first domino falls

fall 1 � �

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Induction

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Inductive hypothesis: Suppose first k dominoes k fall.

Inductive conclusion: Domino k+1 falls.

Base case: The first domino falls.

Basic structure of induction proof

Claim: � � forall� � 0, 1, 2, 3…Base case: ��0� is true.

Inductive step:

��. � 0 , � 1 ,…� � → ��� � 1�

��.��0�� � �. � � � → ��� � 1�

.

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Strong Induction

Inductive hypothesis

Inductive conclusion

Claim: For any natural integer �, �� � � is divisible by 3.

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Claim: For any natural integer �, �� � � is divisible by 3.

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Achilles and the tortoise paradox

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Induction doesn’t quite work over ��

If I show P(0) holds and for every � � 0 there is

some � such that P holds for all elements y in

the range ��, � � �), then it need not be true

that P holds for all non­negative reals.

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Example

� � : � � 2Incorrect proof that

for every � ∈ ���, � � holds.

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Induction on

• Induction is particular to structures like �

• In particular discrete finite structures admit induction (as we can talk about the number of elements in the structures and map it to �).

• Induction: particularly useful for proving properties of discrete finite structures

• Recursion: particularly useful for defining functions on discrete finite structures

(and hence gives a basis for all computation)17

Later…

• Induction on well­ordered sets.

• Induction and its relation to a proof by

contradiction.

• ``Weak’’ and ``strong’’ induction

• Induction on discrete structures– graphs,

trees, words, lists, etc.

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