Section 5.1: Proofs by contradiction (indirect proofs)
What is the proof by contradiction that √2 is irrational?
1. Either √2 is rational or irrational 2. Assume √2 is a rational number --that is, it can be written in the form of a/b, where a and b are coprime integers (share nothing in common except 1) and b ≠0 3. so, we see √2 = a/b. Squaring both sides we obtain 2 = a²/b² 4. Multiplying both sides by b², we obtain that a²=2b² 5. By definition, this means a² is an even number 6. However, we know that if a² is even, then so is a. 7. Thus we can write: a = 2c where c is some integer (this is the definition of an even number) 8. Plugging a = 2c back into our bold equation, we see (2c)² = 2b² ⟹4 c² = 2b² ⟹ b² = 2c² 9. However, this implies that b² is also an even number which means that b is also an even number 10. This means that a and b are both even numbers, which means their greatest common factor is at least 2!! 11: STOP: Orignally we stated that a and b are coprime integers (their GCF is 1) ..CONTRADICTION 12. THUS, due to this impossibility, we conclude √2 is irrational
What are the steps for: Proof by Contradiction/Indirect Proof Procedure?
1. List the possibilities for the conclusion 2. Assume that the negation of the desired conclusion is correct. NOT the negation of the given! 3. Write a chain of reasoning until you reach a contradiction/impossibility. This will be contradiction of either: a. Given information b. A theorem, definition, or another known fact. 4. State the remaining possibility
What are the common number theories?
1. The √2 is irrational 2. there are infinitely many prime numbers
What is the summary of proofs by contradiction (indirect proof)
1. You must prove⟹q 2. You assume p and ?? q 3. You go through your chain of reasoning 4. You reach a contradiction (tells you the assumption is wrong) and realize q must be the actual conclusion
What is the assumption when you say: contradiction that √2 is irrational start with an assumption
Assume there are finite number of prime numbers
See another proof by contradiction example
See another proof by contradiction example
See the summary proofs by contradiction (indirect proof steps snap shot)
See the summary proofs by contradiction (indirect proof steps snap shot)
What is a direct proof?
This is when you start with your hypothesis (Given) and work your way forwards toward your conclusion(prove statement).
True or false: is this a basic example of proof by contradiction: Statement: If the store was robbed last night, then I am innocent. Proof by contradiction: Assume the store was robbed last night, and I am not innocent(guilty) Then I had to have been at the store at the time it was robbed. However, I was on a plane coming back from Austrailia last nighr, since I really wanted to see some kangroos..it was on my bucket list. Therefore, I was at two locations at the same time.-on a plane and at the store. STOP: Contradiction: This is impossible since you can't be at two locations at the same time. Therefore, our original assumption that I am not innocent must be false. Therefore, we conclude that I am innocent.
True
Does the proof by contradiction that √2 is irrational start with an assumption? Yes or no?
Yes
Are the statements below true or false: The common number theories are: 1. The √2 is very confusioning 2. there are infinitely many prime numbers
fasle
Another word for hypothesis?
given
What is the definition of an even number?
n = 2k for some k∈ℵ
What is another way of stating Proof by contradiction/indirect proof procedure?
p ⇒ q: 1. q, -q 2. assume P and -q 3 3. Write a chain of reasoning until you reach a contradiction/impossibility. This will be contradiction of either: a. Given information b. A theorem, definition, or another known fact. 4. q must be conclusion
Another word for conclusion?
prove statement
see examples--read --learn for Indirect proof
see examples--read --learn for Indirect proof
see examples--read --learn for Indirect proof (proof by contradiction)
see examples--read --learn for Indirect proof (proof by contradiction)
Are the statements below true or false: The common number theories are: 1. The √2 is irrational 2. there are infinitely many prime numbers
true
True or false: Many proofs by contradiction in math come from the field of number theory.
true
True or false: Sometimes it's easier to prove statements by assuming the oppositte of what you're trying to prove and arriving at a contradiction..a statement you arrive at that makes you say: contradiction are everywhere your original statement must have been true.
true