How to construct a proof?

  • B
  • Thread starter PhysicsBoyMan
  • Start date
  • Tags
    Proof
In summary, the conversation discusses proving the statement 2ab <= a2 + b2 for all integers a and b. The participants consider using mathematical induction, playing with equations involving squares, and using the fact that squares are always positive. They eventually come to the conclusion that starting with a separate true statement and transforming it into the desired statement is the simplest proof, and that a proof by contradiction is also possible.
  • #1
PhysicsBoyMan
33
0
a and b are integers

Prove that:
2ab <= a2 + b2

I have tested various values for a and b and determined that the statement seems to be generally true. I'm having a hard time though constructing a formal proof.

It will not do to suppose the statement is wrong and then provide a counterexample. This would only prove that the statement is true for those particular values of a and b, and not for all values.

I learned about mathematical induction but I think that method is used to prove statements about a well-ordered set. a and b can be any integer so I don't think that would work.

I wondered what kind of statement may have simplified to produce a2 + b2 so I played with (a+b) * (a+b) which produced a2 + b2 + 2ab and also with (a-b) * (a-b) which produced a2 + b2 - 2ab. This looks similar to the original statement, and I feel like I may be onto some clues.

Without giving me the answer, I wonder if someone could give me a push in the right direction?
 
Physics news on Phys.org
  • #2
The last formula gives you the answer. What do you know about squares?
 
  • #3
fresh_42 said:
The last formula gives you the answer. What do you know about squares?
Well I know that squares are always positive.
 
  • #4
Then gather all together. Start with "all square are positive". Then substitute a certain square of one of your formulas and rearrange the terms..
 
  • #5
fresh_42 said:
Then gather all together. Start with "all square are positive". Then substitute a certain square of one of your formulas and rearrange the terms..
I don't know what some of your comment means. I'm trying hard to figure it out. I'm looking at the term (a-b)2 and I think that may be the "certain square" you are talking about. It's equal to a2 + b2 - 2ab though so I don't know what I could substitute it for. Maybe I'm on the wrong track here altogether.
 
  • #6
PhysicsBoyMan said:
I don't know what some of your comment means. I'm trying hard to figure it out. I'm looking at the term (a-b)2 and I think that may be the "certain square" you are talking about. It's equal to a2 + b2 - 2ab though so I don't know what I could substitute it for. Maybe I'm on the wrong track here altogether.
No, you are done. The square is not negative (##≥ 0##) and all which remains, is to add ##2ab## on both sides.
 
  • #7
fresh_42 said:
No, you are done. The square is not negative (##≥ 0##) and all which remains, is to add ##2ab## on both sides.
How is that a proof that 2ab is always less than or equal to a2 + b2 when a and b are integers?
 
  • #8
PhysicsBoyMan said:
How is that a proof that 2ab is always less than or equal to a2 + b2 when a and b are integers?
The only way to help you further is to type in the proof.

Because you can only derive true statements from true statements, it is a proof:
##0 ≤ (a - b)^2## for all integers, and even for real numbers, too, is a true statement. The next step is to multiply the square which is of course also true. Shifting numbers by addition doesn't change the order, so you are allowed to add ##2ab## on both sides and get again a true statement, which is the one you want to have.
 
  • Like
Likes PhysicsBoyMan
  • #9
Thanks. I didn't think about it like that. That is, starting with a separate true statement and transforming it into the one I want.
 
  • #10
PhysicsBoyMan said:
Thanks. I didn't think about it like that. That is, starting with a separate true statement and transforming it into the one I want.
That's probably the simplest proof but you could have done it simply by contradiction as well:

Suppose ## a^2 + b^2 < 2ab ## then

##a^2+b^2 -2ab <0##
And
##(a-b)^2 <0##

Which is a contradiction.
 
  • Like
Likes PhysicsBoyMan
  • #11
d
PeroK said:
That's probably the simplest proof but you could have done it simply by contradiction as well:

Suppose ## a^2 + b^2 < 2ab ## then

##a^2+b^2 -2ab <0##
And
##(a-b)^2 <0##

Which is a contradiction.
Nice one Perok. Contradictions are a lot more intuitive often times.
 

1. What is a proof?

A proof is a logical and rigorous argument that demonstrates the truth of a statement or proposition. It is used in mathematics and other sciences to establish the validity of a claim.

2. What are the steps to construct a proof?

The steps to construct a proof can vary, but generally include: 1) understanding the statement to be proven, 2) identifying any given information or assumptions, 3) choosing a method or approach for the proof, 4) writing out the logical steps of the proof, and 5) concluding with a restatement of the original statement and how it was proven.

3. What are common methods used to construct a proof?

Some common methods used to construct a proof include direct proof, proof by contradiction, proof by induction, and proof by contrapositive. Each method has its own set of steps and techniques for constructing a proof.

4. How do I know if my proof is correct?

To ensure the correctness of a proof, it is important to carefully follow the logical steps and make sure they are valid. It can also be helpful to have someone else review the proof and check for any errors or gaps in reasoning.

5. Can a proof be written in different ways?

Yes, there are often multiple ways to construct a proof for a given statement. Some methods may be more straightforward or elegant than others, but as long as the proof is logically sound and follows the necessary steps, it can be considered valid.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
704
  • Engineering and Comp Sci Homework Help
Replies
7
Views
819
  • Precalculus Mathematics Homework Help
Replies
6
Views
1K
  • Quantum Physics
Replies
32
Views
2K
Replies
13
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
746
  • Set Theory, Logic, Probability, Statistics
2
Replies
40
Views
6K
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
Back
Top