How to prove that a result does not hold in general

  • Thread starter ver_mathstats
  • Start date
  • Tags
    General
In summary, the statement "For all integers a, b, and c, if a divides c, b divides c, and (a,b)=1, then ab divides c" can be proven by using the fact that there exist integers x and y such that ax + by = 1, which leads to cax + cby = c. However, the statement does not hold in general when (a,b)≠1, and it only takes a single counter-example to prove this. For example, if (a,b)=3, we can find specific values for a, b, and c to show that the statement is false.
  • #1
ver_mathstats
260
21

Homework Statement


Prove that for all a, b, c ∈ ℤ, if a|c and b|c and (a,b)=1 then ab|c. Prove result does not hold in general when (a,b)≠1.

Homework Equations

The Attempt at a Solution


This is not my formal proof it's just the scratch work, for the first part, I have there are integers x and y such that ax + by = 1 which leads us to cax + cby = c. Since a|c and b|c there are integers f and g such that c=fa and c=gb which leads us to gbax + faby = c. Since ab divides the left side it must mean it divides the right side. I am just confused how to prove the second part of the question. I thought instead I could do (a,b)=3 and show that it does not work but I do not think that is right. Any suggestions would be appreciated.

Thank you.
 
Physics news on Phys.org
  • #2
It only takes a single counter-example to prove the second part.
 
  • #3
Looks good so far. You can work with ##(a,b)=3## as an example. Just find specific numbers ##a,b,c## so that the statement is false.
 
  • #4
phyzguy said:
It only takes a single counter-example to prove the second part.
Thank you, I got it now.
 
  • #5
fresh_42 said:
Looks good so far. You can work with ##(a,b)=3## as an example. Just find specific numbers ##a,b,c## so that the statement is false.
Thank you for the reply, I did find values for a, b, and c.
 

Related to How to prove that a result does not hold in general

1. How can I prove that a result does not hold in general?

To prove that a result does not hold in general, you can use a counterexample. This means finding a specific case where the result does not hold and demonstrating why it does not work. This counterexample should be applicable to all cases, not just a few specific ones.

2. What is the importance of proving that a result does not hold in general?

Proving that a result does not hold in general is important because it helps to identify the limitations and boundaries of a theory or concept. It also allows for further research and refinement of the theory to make it more accurate and applicable.

3. Can I use mathematical proofs to show that a result does not hold in general?

Yes, mathematical proofs can be used to demonstrate that a result does not hold in general. This involves using logical reasoning and mathematical principles to show that the result is not universally true.

4. Are there any other methods besides counterexamples to prove that a result does not hold in general?

Yes, besides counterexamples, you can also use proof by contradiction or proof by induction to show that a result does not hold in general. These methods involve assuming the opposite of the result and then showing that it leads to a contradiction or does not hold for all cases.

5. How can I ensure that my proof that a result does not hold in general is valid?

To ensure the validity of your proof, you should carefully define your terms and assumptions, clearly state your reasoning and steps, and check for any errors or inconsistencies. It is also helpful to have your proof reviewed by other experts in the field to ensure its accuracy.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
435
  • Calculus and Beyond Homework Help
Replies
3
Views
609
  • Calculus and Beyond Homework Help
Replies
2
Views
308
  • Calculus and Beyond Homework Help
Replies
2
Views
730
  • Calculus and Beyond Homework Help
Replies
16
Views
4K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
16
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
Back
Top