Proof by counterexample (HSC SSCE Mathematics Extension 2): Flashcards
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What is proof by counterexample?
What is proof by counterexample?
A method to disprove a statement using one example.
What does a 'for all' statement mean?
What does a 'for all' statement mean?
It asserts a property holds for every element in a set.
How do you disprove a 'for all' statement?
How do you disprove a 'for all' statement?
Find a counterexample that does not satisfy the property.
Provide an example of an even number not divisible by 4.
Provide an example of an even number not divisible by 4.
The number 6 is even but not divisible by 4.
What is a counterexample?
What is a counterexample?
A specific case that shows a statement is false.
Counterexample for prime numbers being odd?
Counterexample for prime numbers being odd?
The number 2 is prime and even, disproving the claim.
What are counterexamples for squares not being greater than 1?
What are counterexamples for squares not being greater than 1?
0 and 1 show and are not greater.
What happens if a counterexample is found?
What happens if a counterexample is found?
It invalidates the entire 'for all' assertion.
Counterexample for n^2 + n + 17 being prime?
Counterexample for n^2 + n + 17 being prime?
For n = 16, it equals 289, which is not prime.
Why are counterexamples important in math?
Why are counterexamples important in math?
They refine claims and improve theorem accuracy.