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I am still struggling to understand exactly when would we do a proof by contradiction vs direct proof. For example, in order to prove the square root of 2 is irrational, we would have to do a proof by contradiction and assume the square root of 2 is rational. I am confused about how we can know to do a proof by contradiction by just looking at it from the beginning.

BlueName321
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    You try one proof method, if it works good. Else, try another proof method. – Michael Dec 07 '22 at 04:14
  • The definition of "irrational" is "not rational." Assuming that a number is rational and deriving a contradiction in order to prove it is irrational is thus not a "proof by contradiction." It is the direct way of proving $¬P$. "Proof by contradiction" involves assuming $¬P$, deriving a contradiction, then concluding $P$. – Dan Doel Dec 07 '22 at 04:28
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    Proof by contradiction is often useful when you are trying to prove that a certain (type of) object does not have a particular property, or is not of a particular form. For example: "$\sqrt{2}$ is not rational" or "the set $\mathbb{R}$ of all real numbers is not countable". – Christian E. Ramirez Dec 07 '22 at 04:29
  • Related: https://math.stackexchange.com/questions/679232/substitute-proof-by-contradiction-negation-with-direct-proof-or-proof-by-contrap – Gerry Myerson Dec 07 '22 at 06:22
  • Also, https://math.stackexchange.com/questions/909386/direct-proof-and-contradicttion and https://math.stackexchange.com/questions/2880196/can-all-proven-theorems-be-proven-by-contradiction – Gerry Myerson Dec 07 '22 at 06:28
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    @DanDoel I disagree : In the usual irrationality proof of $\sqrt{2}$ , we assume that it can be represented as $\frac{a}{b}$ with integers $a,b$ (that $\sqrt{2}$ is rational) so we assume the opposite and show that this leads to a contradiction. A clear contradiction proof in my eyes. – Peter Dec 07 '22 at 07:32
  • There is no general way to find the suitable strategy to prove something. – Peter Dec 07 '22 at 07:36
  • In general, there are two types of statements (at least in first order logic). Statements of the form $\exists x(\phi (x))$ (existential statements), and of $\forall x(\phi (x))$ (universal statements). Typically, it is easier in daily life to prove something is true for a single object, than for all of them. In this way, if we are proving something like $\exists x(\phi (x))$, then a direct prove is (likely) easier. If we are proving something like $\forall x(\phi (x))$ then a proof by contradiction is going to be easier, since assuming the contrary turns it into an existential statement again – Graviton Dec 07 '22 at 09:10
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    If a theorem is of the form "this doesn't exist", use proof by contradiction; if it's of the form "this does exist", find an example. (Theorems of the form "for every X there's a Y such that..." are subtler because, although they can be rephrased as "there's no X for which there's no Y such that...", it's usually best to not use contradiction when proving them.) I'm stepping somewhat on @Graviton's toes here. – J.G. Dec 07 '22 at 09:41
  • Does this answer your question? Logical limitations of Proofs by Contradiction – user95921 Dec 26 '23 at 06:41

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In general, it is often adequate to do a proof by contradiction whenever you are supposed to prove that a certain claim always holds and you already know or say suspect that there is at least one exception to it. However, it can also be used to show that a particular claim is always true. That is, you would show that in case the property does in fact not hold in a specific case, it then does effectively lead to a contradiction, hence it must hold under any condition.

Dylan Schmidt
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