Proof by contradiction

In logic, proof by contradiction is a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition. It starts by positing the opposite proposition is true, and then shows that such an assumption leads to a contradiction. Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem. It is a particular kind of the more general form of argument known as reductio ad absurdum.^[1]^[2]

G. H. Hardy described proof by contradiction as "one of a mathematician's finest weapons", saying "It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." ^[1]

Principle

Proof by contradiction is based on the law of noncontradiction as first formalized as a metaphysical principle by Aristotle. Noncontradiction is also a theorem in propositional logic. This states that an assertion or mathematical statement cannot be both true and false. That is, a proposition Q and its negation ~Q ("not-Q") cannot both be true. In a proof by contradiction it is shown that the denial of the statement being proved results in such a contradiction. It has the form of a reductio ad absurdum argument. If P is the proposition to be proved:

P is assumed to be false, that is ~P is true.
It is shown that ~P implies two mutually contradictory assertions, Q and ~Q.
Since these cannot both be true, the assumption that P is false must be wrong, and P must be true.

An alternate form derives a contradiction with the statement to be proved itself:

P is assumed to be false.
It is shown that ~P implies P.
Since P and ~P cannot both be true, the assumption must be wrong and P must be true.

Examples

Irrationality of the square root of 2

A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational.^[3] If it were rational, it could be expressed as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. But if a/b = √2, then a² = 2b². Therefore a² must be even. Because the square of an odd number is odd, that in turn implies that a is even. This means that b must be odd because a/b is in lowest terms.

On the other hand, if a is even, then a² is a multiple of 4. If a² is a multiple of 4 and a² = 2b², then 2b² is a multiple of 4, and therefore b² is even, and so is b.

So b is odd and even, a contradiction. Therefore the initial assumption—that √2 can be expressed as a fraction—must be false.

The length of the hypotenuse

The method of proof by contradiction has also been used to show that for any non-degenerate right triangle, the length of the hypotenuse is less than the sum of the lengths of the two remaining sides.^[4] The proof relies on the Pythagorean theorem. Letting c be the length of the hypotenuse and a and b the lengths of the legs, the claim is that a + b > c.

The claim is negated to assume that a + b ≤ c. Squaring both sides results in (a + b)² ≤ c² or, equivalently, a² + 2ab + b² ≤ c². A triangle is non-degenerate if each edge has positive length, so it may be assumed that a and b are greater than 0. Therefore, a² + b² < a² + 2ab + b² ≤ c². The transitive relation may be reduced to a² + b² < c². It is known from the Pythagorean theorem that a² + b² = c². This results in a contradiction since strict inequality and equality are mutually exclusive. The latter was a result of the Pythagorean theorem and the former the assumption that a + b ≤ c. The contradiction means that it is impossible for both to be true and it is known that the Pythagorean theorem holds. It follows that the assumption that a + b ≤ c must be false and hence a + b > c, proving the claim.

No least positive rational number

Consider the proposition, P: "there is no smallest rational number greater than 0". In a proof by contradiction, we start by assuming the opposite, ¬P: that there is a smallest rational number, say, r.

Now r/2 is a rational number greater than 0 and smaller than r. (In the above symbolic argument, "r/2 is the smallest rational number" would be Q and "r (which is different from r/2) is the smallest rational number" would be ¬Q.) But that contradicts our initial assumption, ¬P, that r was the smallest rational number. So we can conclude that the original proposition, P, must be true — "there is no smallest rational number greater than 0".

Other

For other examples, see proof that the square root of 2 is not rational (where indirect proofs different from the above one can be found) and Cantor's diagonal argument.

In mathematical logic

In mathematical logic, the proof by contradiction is represented as:

S \cup \{ P \} \vdash \mathbb{F}

then

S \vdash \neg P.

S \cup \{ \neg P \} \vdash \mathbb{F}

then

S \vdash P.

In the above, P is the proposition we wish to prove, and S is a set of statements, which are the premises—these could be, for example, the axioms of the theory we are working in, or earlier theorems we can build upon. We consider P, or the negation of P, in addition to S; if this leads to a logical contradiction F, then we can conclude that the statements in S lead to the negation of P, or P itself, respectively.

Note that the set-theoretic union, in some contexts closely related to logical disjunction (or), is used here for sets of statements in such a way that it is more related to logical conjunction (and).

A particular kind of indirect proof assumes that some object doesn't exist, and then proves that this would lead to a contradiction; thus, such an object must exist. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of argument as universally valid. See further Nonconstructive proof.

Notation

Proofs by contradiction sometimes end with the word "Contradiction!". Isaac Barrow and Baermann used the notation Q.E.A., for "quod est absurdum" ("which is absurd"), along the lines of Q.E.D., but this notation is rarely used today.^[5] A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley.^[6] Others sometimes used include a pair of opposing arrows (as $\rightarrow\!\leftarrow$ or $\Rightarrow\!\Leftarrow$ ), struck-out arrows ( $\nleftrightarrow$ ), a stylized form of hash (such as U+2A33: ⨳), or the "reference mark" (U+203B: ※).^[7]^[8] The "up tack" symbol (U+22A5: ⊥) used by philosophers and logicians (see contradiction) also appears, but is often avoided due to its usage for orthogonality.

References

1 2 G. H. Hardy, A Mathematician's Apology; Cambridge University Press, 1992. ISBN 9780521427067. PDF p.19.
↑ S. M. Cohen, "Introduction to Logic", Chapter 5 "proof by contradiction ... Also called indirect proof or reductio ad absurdum ..."
↑ Alfeld, Peter (16 August 1996). "Why is the square root of 2 irrational?". Understanding Mathematics, a study guide. Department of Mathematics, University of Utah. Retrieved 6 February 2013.
↑ Stone, Peter. "Logic, Sets, and Functions: Honors" (PDF). Course materials. pp 14–23: Department of Computer Sciences, The University of Texas at Austin. Retrieved 6 February 2013.
↑ "Math Forum Discussions".
↑ B. Davey and H.A. Priestley, Introduction to lattices and order, Cambridge University Press, 2002.
↑ The Comprehensive LaTeX Symbol List, pg. 20. http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf
↑ Gary Hardegree, Introduction to Modal Logic, Chapter 2, pg. II–2. http://wayback.archive.org/web/20110607061046/http://people.umass.edu/gmhwww/511/pdf/c02.pdf

External links

Proof by Contradiction from Larry W. Cusick's How To Write Proofs

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