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In mathematics it is not possible to derive something wrong when my premise is true. So my question is, is this the reason when i do have a contradiction that i assume my premise had to be wrong, since then it is true again. So we always want true assumptions ?

  • as an example i can say that f differntiable => it is continous. And not the other way around. So if i assume when f is continous and want to show that is differentiable is this not possible. Thus f continous implies f differentiable is wrong ? –  Dec 19 '18 at 18:15
  • so wouldnt i have a contradiction If f is continous i don't have that f is differentiable then i could say f is continous would be wrong hence f would not be continous ?! but f is continous ? –  Dec 19 '18 at 18:19
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    Suppose you go like $A$ implies $B$ implies $C$ with each step correct, and say $C$ is wrong. Since every step you followed was correct, the only possible way the contradiction occurred because your premise was wrong(all other steps were correct). – 1.414212 Dec 19 '18 at 18:20
  • See also the post Justification of Proofs by Contradiction. – Mauro ALLEGRANZA Dec 20 '18 at 08:10

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I don't understand your comments. But a proof by contradiction usually goes like this: You suppose something is true, then show that it is wrong. That, or you give a counterexample.

For example, a continuous function is not necessarily differentiable. Take $f(x) = |x|$ as a counterexample.

A well-used example is the irrationality of $\sqrt 2$.

kmini
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  • yeah exactly. But i assume f is continous and show that f is not differntiable. Would that be a contradiction ? Hence i can say f is continous would be wrong but this is not the case –  Dec 19 '18 at 18:23
  • @MahtsGuy That's assuming your premise is true in the first place. – kmini Dec 19 '18 at 18:24
  • i will try again to explain that: So i assume f is continous and want to show f is differentiable(this not true) this is not possible hence i have a contradiction and therefore f is continous must be wrong? But f can be continous –  Dec 19 '18 at 18:26
  • No, the entire statement $f$ continuous $\Rightarrow f$ differentiable is false. – kmini Dec 19 '18 at 18:28
  • i know but why ? –  Dec 19 '18 at 18:28
  • "But i assume f is continuous and show that f is not differentiable." I don't understand what you mean by that. Some continuous functions are differentiable. If you mean you have a specific continuous function and show it is not differentiable (for example f(x)= |x|) that is not "contradicting" anything. – user247327 Dec 19 '18 at 18:28
  • To be clear: f is continous $\Rightarrow $ f is differentiable. Is wrong (I KNOW). But I assume f is continous and show that f is not differentiable. So i do have a contradiction or dont i? and thus my premise would be wrong. But this cannot be the case –  Dec 19 '18 at 18:31
  • exactly what @user247327 said, there are quantifiers. When you say continuity implies differentiability, this is a statement saying that $\textbf{all}$ continuous functions are differentiable. – kmini Dec 19 '18 at 18:33
  • If you assume that(your premise) all $f$ are continuous(meaning, $f$ can be anything) then you cannot actually show that they are not differentiable. – 1.414212 Dec 19 '18 at 18:38
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There are two implications. a sure and a not sure.

If $P$ is true and the implication $P\implies Q$ is sure, then we are sure that $Q$ is true.

If $P$ is true and the implication $P\implies Q$ is not sure, then we can say nothing about $Q$.

If the implication is sure and $Q$ is false, this means that $P$ is false.

hamam_Abdallah
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