Mathematical Proofs

A mathematical proof is an inferential argument for a mathematical statement]], showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Methods of proof

Direct proof

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove 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 some integers a and b. Then the sum is 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 the distributive property.

Proof by mathematical induction

Despite its name, mathematical induction is a method of deduction, not a form of inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. This avoids having to prove each case individually. A variant of mathematical induction is proof by infinite descent, which can be used, for example, 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: Let N = {1, 2, 3, 4, ...} be the set of natural numbers, and let 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 positive integers of the form 2n − 1 are odd. Let P(n) represent "2n − 1 is odd":

(i) For n = 1, 2n − 1 = 2(1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.

(ii) For any n, if 2n − 1 is odd (P(n)), then (2n − 1) + 2 must also be odd, because adding 2 to an odd number results in an odd number. But (2n − 1) + 2 = 2n + 1 = 2(n+1) − 1, so 2(n+1) − 1 is odd (P(n+1)). So P(n) implies P(n+1).

Thus 2n − 1 is odd, for all positive integers n.

The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".

Proof by contraposition

Proof by contraposition infers the statement "if p then q" by establishing the logically equivalent contrapositive statement: "if not q then not p".

For example, contraposition can be used to establish that, given an integer $x$, if $x^{2}$ is even, then $x$ is even:

Suppose $x$ is not even. Then $x$ is odd. The product of two odd numbers is odd, hence $x^{2}=x\cdot x$ is odd. Thus $x^{2}$ is not even. Thus, if $x^{2}$ is even, the supposition must be false, so $x$ has to be even.

Proof by contradiction

In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that ${\sqrt {2}}$ is an irrational number:

Suppose that ${\sqrt {2}}$ were a rational number. Then it could be written in lowest terms as ${\sqrt {2}}={a \over b}$ 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 expression on the left, 2 must also divide the equal expression on the right. That is, a² is even, which implies that a must also be even, as seen in the proposition above (in Proof by contraposition). 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 have 2 as a common factor. This contradicts our previous statement that a and b have no common factor, so we must conclude that ${\sqrt {2}}$ is an irrational number.

To paraphrase: if one could write ${\sqrt {2}}$ as a fraction, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.

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

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 still has over 600 cases.

Probabilistic proof

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 prove existence theorems.

In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does not specify which candidates have the property, but the probability could not be positive without at least one.

A probabilistic proof 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. While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's probabilistic algorithm for testing primality) are as good as genuine mathematical proofs.

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

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 proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following 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 number of the form $a^{b}.$

Resources

• Mathematical proof, Wikipedia
• Course Textbook, pages 47-53

Cupillari, p. 20.