Proof by Contradiction
Discrete Structures (CS 173)
Madhusudan Parthasarathy, University of Illinois
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Proof by Contradiction
• Sometimes you want to show that something is impossible
– 2 cannot be written as a ratio of integers
– There is no compression algorithm that reduces the size of all files
– A cycle with an odd number of nodes can’t be colored with two colors
• Difficult to prove non‐existence directly, and can’t prove by example
• Solution: show that the negation of the claim leads to a contradiction
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Contradiction
A set of propositions is a contradiction if their conjunction is always false
Contradiction?
∧
∨
5 ∧
21
20 and is odd
5 ∧
21
5 ∧
21
is negative number and is real
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Proof by contradiction
Claim: There are infinitely many prime numbers
Equivalent claim: There is not a finite set of primes.
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Basic form of proof by contradiction
1. I need to show proposition 2. Suppose, instead, that is false.
3. Then, show that both and
a contradiction.
are true, which is 4. Therefore, our assumption that p is false must
be impossible.
So must be true.
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Structure of a proof by contradiction
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Why proof by contradiction works
We need to prove proposition Instead, we show , i.e., that we can conclude a contradiction from not By contrapositive, 7
Another explanation
We need to prove proposition Instead, we show , i.e., that we can conclude a contradiction from But means that , so 8
Danger of proof by contradiction: a mistake in the proof might also lead to a contradiction
See this blog post about P=NP problem
http://rjlipton.wordpress.com/2011/01/08/proofs‐by‐contradiction‐and‐other‐dangers/
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Proof by contradiction
Claim: 2 is irrational
Equivalent claim: There does not exist a pair of integers ,
without common factors such that 2
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Another proof…
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Proof by contradiction
There re are no positive integer solutions to the diophantine
equation 1
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Proof by contradiction
There are no rational number solutions to the equation 1 0
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Proof by contradiction
If a is a rational number and b is an irrational number, then a+b is an irrational number.
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Proof by contradiction
Claim: No lossless compression algorithm can reduce the size of every file.
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Why image compression works: images are mostly smooth
Lossless compression (PNG)
1. Predict that a pixel’s value based on its upper‐left neighborhood
2. Store difference of predicted and actual value
3. Pkzip it (DEFLATE algorithm)
Proof by contradiction
Claim: A cycle graph with an odd number of nodes is not 2‐
colorable.
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Proof by contradiction vs contrapositive
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When to use each type of proof
Match the situation to the proof type
Situation
1.
2.
3.
Can see how conclusion directly follows from hypothesis
Need to demonstrate claim for an unbounded set of integers
Proof type
a)
Direct proof
b) Proof by example or counter‐example
Easier to show that negation of hypothesis follows from negation of conclusion
c)
4.
Need to show that something doesn’t exist
d) Induction
5.
Need to show that something exists
Proof by contrapositive (or logical equivalence)
e) Proof by contradiction
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When to use each type of proof
Match the situation to the proof type
Situation
1.
2.
3.
Can see how conclusion directly follows from claim (a)
Need to demonstrate claim for an unbounded set of integers (d)
Proof type
a)
Direct proof
b) Proof by example or counter‐example
Easier to show that negation of hypothesis follows from negation of conclusion (c)
c)
4.
Need to show that something doesn’t exist (e)
d) Induction
5.
Need to show that something exists (b)
Proof by contrapositive (or logical equivalence)
e) Proof by contradiction
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Later in this course
• Prove that there is no bijection between natural numbers and reals
(not even a 1‐1 function from reals to nat)
• Prove that there is no bijection between S and P(S).
• Prove that there are problems that no computer can solve.
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