Assumptions

[JulianMathHelp]

Are we allowed to use assumptions in the start of proofs/math problems as long as in our further work, we disprove (in cases of proofs) or prove our assumption? Coming up with the final answer, with a proved or disproved assumption?

 

[lev888]

Are we allowed to use assumptions in the start of proofs/math problems as long as in our further work, we disprove (in cases of proofs) or prove our assumption? Coming up with the final answer, with a proved or disproved assumption?

Please provide an example. I would say no. Order matters, it ensures we don't end up with a circular argument - you make an assumption A and use it to prove a theorem T, and then you use T to prove A.

 

[JulianMathHelp]

I was thinking about it, and the only case where an assumption would work if proving an assumption wrong right?

 

[lev888]

I was thinking about it, and the only case where an assumption would work if proving an assumption wrong right?

Are you referring to proofs by contradiction? I wouldn't describe them as cases where "an assumption would work". But, yes, we make an assumption, if it leads to a contradiction, then the negation of the assumption is true.

 

[JulianMathHelp]

Are you referring to proofs by contradiction? I wouldn't describe them as cases where "an assumption would work". But, yes, we make an assumption, if it leads to a contradiction, then the negation of the assumption is true.

Yes, so using an assumption to prove it wrong would work, but I don’t think I know any others where assumptions would help me find an answer.

 

[Steven G]

Here is a great example using the technique you mention.

Fact: Given any non-empty set A, { } is a subset of A

Proof: Suppose { } is NOT a subset of A. Then there must be an element in the empty set that is not in A. But this is absurd as the empty set has no elements. Therefore our assumption that { } is not a subset of A is is wrong. So we must conclude that { } is a subset of A.

 

[JulianMathHelp]

Here is a great example using the technique you mention.

Fact: Given any non-empty set A, { } is a subset of A

Proof: Suppose { } is NOT a subset of A. Then there must be an element in the empty set that is not in A. But this is absurd as the empty set has no elements. Therefore our assumption that { } is not a subset of A is is wrong. So we must conclude that { } is a subset of A.

So the thing we are trying to prove is “Given any non-empty set A, { } is a subset of A“ So the best strategy is to assume the opposite (which we are allowed to do as we currently have no proof that our assumption is automatically wrong), and then prove that our assumption is incorrect, meaning the the thing we are trying to prove is true. Is this how it goes? Thanks for the example!

 

[Steven G]

So the thing we are trying to prove is “Given any non-empty set A, { } is a subset of A“ So the best strategy is to assume the opposite (which we are allowed to do as we currently have no proof that our assumption is automatically wrong), and then prove that our assumption is incorrect, meaning the the thing we are trying to prove is true. Is this how it goes? Thanks for the example!

You seem to understand it correctly. As far as my proof being the best strategy is a matter of opinion.