Verification of simple inequality proof

In summary, the conversation discusses two different proofs for the inequality a < (a+b)/2 < b, with one being from a book and the other being the reader's own. The reader asks if their proof is also valid and is given feedback on the direction of implications used in their proof. The conversation concludes by mentioning that the method used in the reader's proof is called "synthetic proof" and is often used in trigonometry.
  • #1
GeoMike
67
0
The question given is:
If a < b, prove that a < (a+b)/2 < b

The book had a different proof than the one I came up with. I understand the book's proof, I just want to know if my proof is also ok.

I did the following:

a < (a+b)/2 < b
2a < a+b < 2b
a < b < 2b-a
a-b < 0 < b-a

Since it was given that a < b, a-b must be less than 0, and b-a must be greater than zero, so the inequality a < (a+b)/2 < b is true if a < b.

Is this ok?

Thanks,
-GeoMike-
 
Last edited:
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  • #2
It looks a bit like you did the proof in reverse. It seems that you started with a<(a+b)/2<b and then got to a<b rather than starting with a<b and proving a<(a+b)/2<b like the problem seems to want.
 
  • #3
really you should show the direction of the implications you're using, so what you mean is actually

[tex]a<(a+b)/2<b \Longleftarrow 2a<(a+b)<2b \Longleftarrow a<b<2b-a \Longleftarrow a-b<0<b-a \Longleftarrow a<b,[/tex]

and not the other way. It doesn't matter much here since all the inequalities there are equivalent (so the implications work both ways, ie. in fact [itex]a<b \Longleftrightarrow a<(a+b)/2<b[/itex]).
 
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  • #4
Proof by reverse is great, but you have to show that the steps can logically be reversed at the end
 
  • #5
That's sometimes called "synthetic proof". It's often used to prove trig identitities. Start with what you want to prove and work back to an obviously true statement. It's valid as long as every stepe is reversible,.
 
  • #6
Thank you for the replies! :smile:

-GeoMike-
 

1. How can I verify if a simple inequality proof is correct?

In order to verify a simple inequality proof, you can follow these steps:

  • Start by identifying the statement that is being proved and the assumptions or given information.
  • Next, carefully examine the logical steps taken by the proof to reach the conclusion.
  • Check if each step follows logically from the previous one and if any assumptions are being used.
  • Make sure that the steps are valid and that no information is being omitted or added.
  • If all the steps are correct and the conclusion follows logically from the assumptions, then the proof is considered valid.

2. What are some common mistakes to look out for when verifying a simple inequality proof?

Some common mistakes to look out for when verifying a simple inequality proof include:

  • Incorrectly applying mathematical rules or properties.
  • Using assumptions that are not given or are not applicable to the statement being proved.
  • Omitting important steps or information.
  • Not clearly stating the logical connections between each step.
  • Using circular reasoning, where the conclusion is used to prove one of the assumptions.

3. Can computer programs be used to verify simple inequality proofs?

Yes, computer programs can be used to verify simple inequality proofs. These programs use formal logic and mathematical rules to check the validity of a proof. However, it is still important for a human to carefully examine the proof to ensure that it is logically sound.

4. What are some tips for writing a clear and valid simple inequality proof?

Here are some tips for writing a clear and valid simple inequality proof:

  • Start by stating the statement that is being proved and the assumptions or given information.
  • Clearly state each step and explicitly state the logical connections between them.
  • Use proper mathematical notation and terminology.
  • Avoid using circular reasoning or making assumptions that are not given or applicable.
  • Check for any errors or omissions before finalizing the proof.

5. Are there any resources or tools available to help with verifying simple inequality proofs?

Yes, there are resources and tools available to help with verifying simple inequality proofs. Some textbooks or online resources provide example proofs and solutions to practice with. Additionally, there are computer programs and online proof checkers that can assist in verifying the validity of a proof.

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