3 ­ Proof by Counter Example.notebook
December 06, 2011
Proofs by
Counter Example
May 19­10:23 AM
May 19­10:23 AM
A conditional statement can be proven false if an example can be found which fits the hypothesis, but the conclusion is false.
Such an example is called a Counter Example.
It only takes one counter example to prove a conditional statement false.
A conditional statement can be proven false if an example can be found which fits the hypothesis, but the conclusion is false.
Such an example is called a Counter Example.
It only takes one counter example to prove a conditional statement false.
Example
Example
if x > 0 then √x < x
if x > 0 then √x < x
May 19­10:35 AM
Why is only one counterexample
enough to disprove a conjecture?
May 19­10:35 AM
Why is only one counterexample
enough to disprove a conjecture?
Pull
Dec 6­1:18 PM
Dec 6­1:18 PM
1
3 ­ Proof by Counter Example.notebook
Conjecture #1
The difference between
consecutive perfect squares
is always an odd number
Conjecture #2
The difference between
consecutive perfect squares
is always a prime number.
December 06, 2011
Conjecture #1
The difference between
consecutive perfect squares
is always an odd number
Conjecture #2
The difference between
consecutive perfect squares
is always a prime number.
How can these conjectures be tested?
Conjecture 1
How can these conjectures be tested?
Pull
Conjecture 2
Apr 19­2:46 PM
Can you find another counterexample to Conjecture #2?
Apr 19­2:46 PM
Can you find another counterexample to Conjecture #2?
Can you find a counterexample to Conjecture #1?
Pull
Can you find a counterexample to Conjecture #1?
Apr 19­2:51 PM
Apr 19­2:51 PM
Matt found an interesting numeric pattern:
1*8+1=9
12 * 8 + 1 = 98
123 * 8 + 1 = 987
1234 * 8 + 1 = 9876
Matt found an interesting numeric pattern:
1*8+1=9
12 * 8 + 1 = 98
123 * 8 + 1 = 987
1234 * 8 + 1 = 9876
Matt thinks that this pattern will continue. Search for a
counterexample to Matt's conjecture.
Matt thinks that this pattern will continue. Search for a
counterexample to Matt's conjecture.
Pull
Apr 30­5:26 PM
Apr 30­5:26 PM
2
3 ­ Proof by Counter Example.notebook
If you had not found a counterexample at the tenth step, should you
have continued looking? When would it be reasonable to stop gathering
evidence if all the examples supported the conjecture? Justify your
decision.
December 06, 2011
If you had not found a counterexample at the tenth step, should you
have continued looking? When would it be reasonable to stop gathering
evidence if all the examples supported the conjecture? Justify your
decision.
Pull
Apr 30­5:28 PM
*
*
Key Ideas
Once you have found a counterexample to a conjecture, you have
disproved the conjecture. This means that the conjecture is
invalid.
You may be able to use a counterexample to help you revise a
conjecture.
Need To Know
*
*
A single counterexample is enough to disprove a conjecture.
Even if you cannot find a counterexample, you cannot be certain that
there is not one. Any supporting evidence you develop while
searching for a counterexample, however, does increase the
likelihood that the conjecture is true.
Apr 30­5:29 PM
Apr 30­5:28 PM
Assignment
Pages 22-24
#1-17
Dec 6­1:18 PM
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