Can something proved by contradiction always be proved without a proof by contradiction?


Can you prove by contradiction?

To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible. That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) – we call this a contradiction.

How do you prove a contradiction is proved?

The steps taken for a proof by contradiction (also called indirect proof) are:

  1. Assume the opposite of your conclusion. …
  2. Use the assumption to derive new consequences until one is the opposite of your premise. …
  3. Conclude that the assumption must be false and that its opposite (your original conclusion) must be true.

Is a contradiction always false?

A contradiction is something that is always false, regardless of it’s truth values.

Is proof by contradiction the same as proof by contrapositive?

In a proof by contrapositive, we actually use a direct proof to prove the contrapositive of the original implication. In a proof by contradiction, we start with the supposition that the implication is false, and use this assumption to derive a contradiction. This would prove that the implication must be true.

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Why does proof by contradiction work?

It’s because a statement can only ever be true or false, there’s nothing in between. The idea behind proof of contradiction is that you basically prove that a hypothesis “cannot be untrue”. I.e., you prove that if the hypothesis is false, then 1=0.

Why is proof by contradiction bad?

One general reason to avoid proof by contradiction is the following. When you prove something by contradiction, all you learn is that the statement you wanted to prove is true. When you prove something directly, you learn every intermediate implication you had to prove along the way.

What is a contradiction statement?

A contradictory statement is one that says two things that cannot both be true. An example: My sister is jealous of me because I’m an only child. Contradictory is related to the verb contradict, which means to say or do the opposite, and contrary, which means to take an opposite view.

Is contradiction and contrapositive same?

The contrapositive says that to argue P⟹Q, you instead argue ∼Q⟹∼P. Argument by contradiction is done by assuming P and showing P⟹False. Proving there is an infinity of primes is done by contradiction.

When should you prove by contradiction?

Contradiction proofs are often used when there is some binary choice between possibilities:

  • 2 \sqrt{2} 2 ​ is either rational or irrational.
  • There are infinitely many primes or there are finitely many primes.

Is proof by contradiction sound?

Proof by contradiction is perfectly sound.” This completely misses the point. There are assumptions that make it sound, and carefully pointing out what they are and why people make them is what the questioner is asking.

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How do you prove by contradiction that there are infinitely many prime numbers?

Proof by contradiction: Assume there are finitely many prime numbers. Then, we can say that there are n prime numbers, and we can write them down, in order: Let 2 = p1 < p2 < … < pn be a list of all the prime numbers. The key trick in the proof is to define the integer N =1+ p1 · p2 · …

What is the method of proof by contradiction Why does this proof method work when would we use this method to prove a statement?

Another method of proof that is frequently used in mathematics is a proof by contradiction. This method is based on the fact that a statement X can only be true or false (and not both). The idea is to prove that the statement X is true by showing that it cannot be false.