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Direct proof Main article: Direct proof In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.[13] For example, direct proof can be used to establish that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for integers a and b. Then the sum x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence the sum of any two even integers is even. This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and distributivity. Proof by mathematical induction Main article: Mathematical induction Mathematical induction is not a form of inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved, which establishes that a certain case implies the next case. Applying the induction rule repeatedly, starting from the independently proved base case, proves many, often infinitely many, other cases.[14] Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. A subset of induction is infinite descent. Infinite descent can be used to prove the irrationality of the square root of two. A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers:[15] Let N = {1,2,3,4,...} be the set of natural numbers, and P(n) be a mathematical statement involving the natural number n belonging to N such that (i) P(1) is true, i.e., P(n) is true for n = 1. (ii) P(n+1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n+1) is true. Then P(n) is true for all natural numbers n. For example, we can prove by induction that all integers of the form 2n + 1 are odd: (i) For n = 1, 2n + 1 = 2(1) + 1 = 3, and 3 is odd. Thus P(1) is true. (ii) For 2n + 1 for some n, 2(n+1) + 1 = (2n+1) + 2. If 2n + 1 is odd, then (2n+1) + 2 must also be odd, because adding 2 to an odd number results in an odd number. So P(n+1) is true if P(n) is true. Thus 2n + 1 is odd, for all natural numbers n. It is common for the phrase "proof by induction" to be used for a "proof by mathematical induction".[16] Proof by contraposition Main article: Contraposition Proof by contraposition infers the conclusion "if p then q" from the premise "if not q then not p". The statement "if not q then not p" is called the contrapositive of the statement "if p then q". For example, contraposition can be used to establish that, given an integer x, if x² is even, then x is even: Suppose x is not even. Then x is odd. The product of two odd numbers is odd, hence x² = x·x is odd. Thus x² is not even. Proof by contradiction Main article: Proof by contradiction In proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction to the absurd"), it is shown that if some statement were true, a logical contradiction occurs, hence the statement must be false. A famous example of proof by contradiction shows that \sqrt{2} is an irrational number: Suppose that \sqrt{2} were a rational number, so by definition \sqrt{2} = {a\over b} where a and b are non-zero integers with no common factor. Thus, b\sqrt{2} = a. Squaring both sides yields 2b2 = a2. Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a2 is even, which implies that a must also be even. So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2 yields b2 = 2c2. But then, by the same argument as before, 2 divides b2, so b must be even. However, if a and b are both even, they share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that \sqrt{2} is an irrational number. Proof by construction Main article: Proof by construction Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example. It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property. Proof by exhaustion Main article: Proof by exhaustion In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four color theorem as of 2011 still has over 600 cases. Probabilistic proof Main article: Probabilistic method A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. Probabilistic proof, like proof by construction, is one of many ways to show existence theorems. This is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work on the Collatz conjecture shows how far plausibility is from genuine proof.[17] Combinatorial proof Main article: Combinatorial proof A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal. Nonconstructive proof Main article: Nonconstructive proof A nonconstructive proof establishes that a mathematical object with a certain property exists without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proven to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. A famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a^b is a rational number: Either \sqrt{2}^{\sqrt{2}} is a rational number and we are done (take a=b=\sqrt{2}), or \sqrt{2}^{\sqrt{2}} is irrational so we can write a=\sqrt{2}^{\sqrt{2}} and b=\sqrt{2}. This then gives \left (\sqrt{2}^{\sqrt{2}}\right )^{\sqrt{2}}=\sqrt{2}^{2}=2, which is thus a rational of the form a^b. Statistical proofs in pure mathematics Main article: Statistical proof The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such as involving cryptography, chaotic series, and probabilistic or analytic number theory.[18][19][20] It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. See also "Statistical proof using data" section below. Computer-assisted proofs Main article: Computer-assisted proof Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.[7] However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight. |
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