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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