Proof by contradiction

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In logic and mathematics proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. It is a particular kind of the more general form of argument known as reductio ad absurdum.

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."

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 ${\displaystyle \lnot }$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:

1. P is assumed to be false, that is ${\displaystyle \lnot }$P is true.
2. It is shown that ${\displaystyle \lnot }$P implies two mutually contradictory assertions, Q and ${\displaystyle \lnot }$Q.
3. Since Q and ${\displaystyle \lnot }$Q 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 by showing that ${\displaystyle \lnot }$P implies P. Thus ${\displaystyle \lnot }$P is false, which implies that P is true. 

An existence proof by contradiction 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 nonconstructive proof as universally valid.

Law of the excluded middle

Proof by contradiction also depends on the law of the excluded middle, also first formulated by Aristotle. This states that either an assertion or its negation must be true

${\displaystyle \forall P\vdash (P\lor \lnot P)}$

(For all propositions P, either P or not-P is true)

That is, there is no other truth value besides "true" and "false" that a proposition can take. Combined with the principle of noncontradiction, this means that exactly one of ${\displaystyle P}$ and ${\displaystyle \lnot P}$ is true. In proof by contradiction, this permits the conclusion that since the possibility of ${\displaystyle \lnot P}$ has been excluded, ${\displaystyle P}$ must be true. 

The law of the excluded middle is accepted in virtually all formal logics; however, some intuitionist mathematicians do not accept it, and thus reject proof by contradiction as a proof technique.

Relationship with other proof techniques

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Proof by contradiction is closely related to proof by contrapositive, and the two are sometimes confused, though they are distinct methods. The main distinction is that a proof by contrapositive applies only to statements ${\displaystyle P}$ that can be written in the form ${\displaystyle A\rightarrow B}$ (i.e., implications), whereas the technique of proof by contradiction applies to statements ${\displaystyle P}$ of any form:

• Proof by contradiction (general): assume ${\displaystyle \lnot P}$ and derive a contradiction.

This corresponds, in the framework of propositional logic, to the equivalence ${\displaystyle P\equiv \lnot \lnot P\equiv \lnot P\to \bot }$, where ${\displaystyle \bot }$ is the logical contradiction, or false value.

In the case where the statement to be proven is an implication ${\displaystyle A\rightarrow B}$, let us look at the differences between direct proof, proof by contrapositive, and proof by contradiction:

• Direct proof: assume ${\displaystyle A}$ and show ${\displaystyle B}$.
• Proof by contrapositive: assume ${\displaystyle \lnot B}$ and show ${\displaystyle \lnot A}$.

This corresponds to the equivalence ${\displaystyle A\rightarrow B\equiv \lnot B\rightarrow \lnot A}$.

• Proof by contradiction: assume ${\displaystyle A}$ and ${\displaystyle \lnot B}$ and derive a contradiction.

This corresponds to the equivalences ${\displaystyle A\rightarrow B\equiv \lnot \lnot (A\rightarrow B)\equiv \lnot (A\rightarrow B)\rightarrow \bot \equiv (A\land \lnot B)\rightarrow \bot }$.

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. 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. 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. But that contradicts our initial assumption, ¬P, that r was the smallest rational number. (In the above symbolic argument, "r is the smallest rational number" would be Q and "r/2 is a rational number smaller than r" would be ¬Q.) 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.

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. A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley. Others sometimes used include a pair of opposing arrows (as ${\displaystyle \rightarrow \!\leftarrow }$ or ${\displaystyle \Rightarrow \!\Leftarrow }$), struck-out arrows (${\displaystyle \nleftrightarrow }$), a stylized form of hash (such as U+2A33: ⨳), or the "reference mark" (U+203B: ※). 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.

Principle of explosion

A curious logical consequence of the principle of non-contradiction is that a contradiction implies any statement; if a contradiction is accepted as true, any proposition (or its negation) can be proved from it. This is known as the principle of explosion (Latin: ex falso quodlibet, "from a falsehood, anything [follows]", or ex contradictione sequitur quodlibet, "from a contradiction, anything follows"), or the principle of pseudo-scotus.

${\displaystyle \forall Q:(P\land \lnot P)\rightarrow Q}$

(for all Q, P and not-P implies Q)

Thus a contradiction in a formal axiomatic system is disastrous; since any theorem can be proven true it destroys the conventional meaning of truth and falsity. 

The discovery of contradictions at the foundations of mathematics at the beginning of the 20th century, such as Russell's paradox, threatened the entire structure of mathematics due to the principle of explosion. This motivated a great deal of work during the 20th century to create consistent axiomatic systems to provide a logical underpinning for mathematics. This has also led a few philosophers such as Newton da Costa, Walter Carnielli and Graham Priest to reject the principle of non-contradiction, giving rise to theories such as paraconsistent logic and dialethism, which accepts that there exist statements that are both true and false.

Euclid's theorem

From Wikipedia, the free encyclopedia

 

Euclid's theorem

Field	Number theory
First proof by	Euclid
First proof in	c. 300 BCE
Generalizations	Dirichlet's theorem on arithmetic progressions Prime number theorem

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of the theorem.

Euclid's proof

Euclid offered a proof published in his work Elements (Book IX, Proposition 20), which is paraphrased here.

Consider any finite list of prime numbers p₁, p₂, ..., p_n. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p₁p₂...p_n. Let q = P + 1. Then q is either prime or not:

• If q is prime, then there is at least one more prime that is not in the list.
• If q is not prime, then some prime factor p divides q. If this factor p were in our list, then it would divide P (since P is the product of every number in the list); but p divides P + 1 = q. If p divides P and q, then p would have to divide the difference of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, p cannot be on the list. This means that at least one more prime number exists beyond those in the list.

This proves that for every finite list of prime numbers there is a prime number not in the list, and therefore there must be infinitely many prime numbers. Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers. While such a proof does follow from Euclid's method, Euclid's proof deduces the infinitude directly.

Variations

Several variations on Euclid's proof exist, including the following:

The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them. Hence, n! + 1 is not divisible by any of the integers from 2 to n, inclusive (it gives a remainder of 1 when divided by each). Hence n! + 1 is either prime or divisible by a prime larger than n. In either case, for every positive integer n, there is at least one prime bigger than n. The conclusion is that the number of primes is infinite.

Euler's proof

Another proof, by the Swiss mathematician Leonhard Euler, relies on the fundamental theorem of arithmetic: that every integer has a unique prime factorization. If P is the set of all prime numbers, Euler wrote that:

${\displaystyle \prod _{p\in P}{\frac {1}{1-{\frac {1}{p}}}}=\prod _{p\in P}\sum _{k\geq 0}{\frac {1}{p^{k}}}=\sum _{n}{\frac {1}{n}}.}$

The first equality is given by the formula for a geometric series in each term of the product. The second equality is a special case of the Euler product formula for the Riemann zeta function. To show this, distribute the product over the sum:

${\displaystyle {\begin{aligned}\prod _{p\in P}\sum _{k\geq 0}{\frac {1}{p^{k}}}&=\sum _{k\geq 0}{\frac {1}{2^{k}}}\times \sum _{k\geq 0}{\frac {1}{3^{k}}}\times \sum _{k\geq 0}{\frac {1}{5^{k}}}\times \sum _{k\geq 0}{\frac {1}{7^{k}}}\times \cdots \\[8pt]&=\sum _{k,\ell ,m,n,\ldots \geq 0}{\frac {1}{2^{k}3^{\ell }5^{m}7^{n}\cdots }}=\sum _{n}{\frac {1}{n}}\end{aligned}}}$

in the result, every product of primes appears exactly once and so by the fundamental theorem of arithmetic the sum is equal to the sum over all integers. 

The sum on the right is the harmonic series, which diverges. Thus the product on the left must also diverge. Since each term of the product is finite, the number of terms must be infinite; therefore, there is an infinite number of primes.

Erdős's proof

Paul Erdős gave a third proof that also relies on the fundamental theorem of arithmetic. First every integer n can be uniquely written as

${\displaystyle rs^{2}}$

where r is square-free, or not divisible by any square numbers (let s² be the largest square number that divides n and then let r = n/s²). Now suppose that there are only finitely many prime numbers and call the number of prime numbers k. As each of the prime numbers factorizes any squarefree number at most once, by the fundamental theorem of arithmetic, there are only 2^k square-free numbers. 

Now fix a positive integer N and consider the integers between 1 and N. Each of these numbers can be written as rs² where r is square-free and s² is a square, like this:

( 1×1, 2×1, 3×1, 1×4, 5×1, 6×1, 7×1, 2×4, 1×9, 10×1, ...)

There are N different numbers in the list. Each of them is made by multiplying a squarefree number, by a square number that is N or less. There are ⌊√N⌋ such square numbers. Then, we form all the possible products of all squares less than N multiplied by all squarefrees everywhere. There are exactly 2^k⌊√N⌋ such numbers, all different, and they include all the numbers in our list and maybe more. Therefore, 2^k⌊√N⌋ ≥ N. Here, ⌊x⌋ denotes the floor function.

Since this inequality does not hold for N sufficiently large, there must be infinitely many primes.

Furstenberg's proof

In the 1950s, Hillel Furstenberg introduced a proof by contradiction using point-set topology.

Define a topology on the integers Z, called the evenly spaced integer topology, by declaring a subset U ⊆ Z to be an open set if and only if it is either the empty set, ∅, or it is a union of arithmetic sequences S(a, b) (for a ≠ 0), where

${\displaystyle S(a,b)=\{an+b\mid n\in \mathbb {Z} \}=a\mathbb {Z} +b.}$

Then a contradiction follows from the property that a finite set of integers cannot be open and the property that the basis sets S(a, b) are both open and closed, since

${\displaystyle \mathbb {Z} \setminus \{-1,+1\}=\bigcup _{p\mathrm {\,prime} }S(p,0)}$

cannot be closed because its complement is finite, but is closed since it is a finite union of closed sets.

Some recent proofs

Proof using the inclusion-exclusion principle

Juan Pablo Pinasco has written the following proof.

Let p₁, ..., p_N be the smallest N primes. Then by the inclusion–exclusion principle, the number of positive integers less than or equal to x that are divisible by one of those primes is

${\displaystyle {\begin{aligned}1+\sum _{i}\left\lfloor {\frac {x}{p_{i}}}\right\rfloor -\sum _{i$

Dividing by x and letting x → ∞ gives 

${\displaystyle \sum _{i}{\frac {1}{p_{i}}}-\sum _{i$

 This can be written as

${\displaystyle 1-\prod _{i=1}^{N}\left(1-{\frac {1}{p_{i}}}\right).\qquad (3)}$

If no other primes than p₁, ..., p_N exist, then the expression in (1) is equal to ${\displaystyle \lfloor x\rfloor }$ and the expression in (2) is equal to 1, but clearly the expression in (3) is not equal to 1. Therefore, there must be more primes than  p₁, ..., p_N.

Proof using de Polignac's formula

In 2010, Junho Peter Whang published the following proof by contradiction. Let k be any positive integer. Then according to de Polignac's formula (actually due to Legendre) 

${\displaystyle k!=\prod _{p{\text{ prime}}}p^{f(p,k)}}$

where

${\displaystyle f(p,k)=\left\lfloor {\frac {k}{p}}\right\rfloor +\left\lfloor {\frac {k}{p^{2}}}\right\rfloor +\cdots .}$

${\displaystyle f(p,k)<{\frac {k}{p}}+{\frac {k}{p^{2}}}+\cdots ={\frac {k}{p-1}}\leq k.}$

But if only finitely many primes exist, then

${\displaystyle \lim _{k\to \infty }{\frac {\left(\prod _{p}p\right)^{k}}{k!}}=0,}$

(the numerator of the fraction would grow singly exponentially while by Stirling's approximation the denominator grows more quickly than singly exponentially), contradicting the fact that for each k the numerator is greater than or equal to the denominator.

Proof by construction

Filip Saidak gave the following proof by construction, which does not use reductio ad absurdum or Euclid's Lemma (that if a prime p divides ab then it must divide a or b). 

Since each natural number (> 1) has at least one prime factor, and two successive numbers n and (n + 1) have no factor in common, the product n(n + 1) has more different prime factors than the number n itself.  So the chain of pronic numbers:

1×2 = 2 {2},    2×3 = 6 {2, 3},    6×7 = 42 {2,3, 7},    42×43 = 1806 {2,3,7, 43},    1806×1807 = 3263443 {2,3,7,43,13,139}, · · ·
 

provides a sequence of unlimited growing sets of primes.

Proof using the irrationality of π

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Representing the Leibniz formula for π as an Euler product gives

${\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}\times {\frac {5}{4}}\times {\frac {7}{8}}\times {\frac {11}{12}}\times {\frac {13}{12}}\times {\frac {17}{16}}\times {\frac {19}{20}}\times {\frac {23}{24}}\times {\frac {29}{28}}\times {\frac {31}{32}}\times \cdots }$

The numerators of this product are the odd prime numbers, and each denominator is the multiple of four nearest to the numerator. 

If there were finitely many primes this formula would show that π is a rational number whose denominator is the product of all multiples of 4 that are one more or less than a prime number, contradicting the fact that π is irrational.

Proof using information theory

Alexander Shen and others have presented a proof that uses incompressibility:

Suppose there were only k primes (p₁... p_k). By the fundamental theorem of arithmetic, any positive integer n could then be represented as:

${\displaystyle n={p_{1}}^{e_{1}}{p_{2}}^{e_{2}}\cdots {p_{k}}^{e_{k}},}$

where the non-negative integer exponents e_i together with the finite-sized list of primes are enough to reconstruct the number. Since ${\displaystyle p_{i}\geq 2}$ for all i, it follows that all ${\displaystyle e_{i}\leq \lg n}$ (where ${\displaystyle \lg }$ denotes the base-2 logarithm). 

This yields an encoding for n of the following size (using big O notation):

${\displaystyle O({\text{prime list size}}+k\lg \lg n)=O(\lg \lg n)}$ bits.

This is a much more efficient encoding than representing n directly in binary, which takes ${\displaystyle N=O(\lg n)}$ bits. An established result in lossless data compression states that one cannot generally compress N bits of information into less than N bits. The representation above violates this by far when n is large enough since ${\displaystyle \lg \lg n=o(\lg n)}$.

Therefore, the number of primes must not be finite.

A generalization: Dirichlet's theorem on arithmetic progressions

Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. Euclid's theorem is a special case of Dirichlet's theorem for a = d = 1. Every case of Dirichlet's theorem yields Euclid's theorem.

A stronger result: the prime number theorem

Let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. The prime number theorem then states that x / log x is a good approximation to π(x), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without bound is 1:

${\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\ln(x)}}=1.}$

Using asymptotic notation this result can be restated as

${\displaystyle \pi (x)\sim {\frac {x}{\log x}}.}$

This yields Euclid's theorem, since ${\displaystyle \lim _{x\rightarrow \infty }{\frac {x}{\log x}}=\infty .}$