Proof (logic)

In general, a proof is a demonstration that a specified statement follows from a set of assumed statements. The specified statement that follows from the assumed statements is called the conclusion of the proof and the assumed statements that the conclusion follows from are called the premises of the proof.

Particularly, in mathematics, a proof is a demonstration that the conlusion is a necessary consequence of the set of premises, i.e. the conclusion must be true if the premises are all true. Also, in logic, a proof is formally meant to be a sequence of formulas in some deductive system that shows the transformation from the set of premises (expresed as formulas) into the conclusion (also expressed as a formula) by the rules specified in the deductive system. The notion of proofs in this sense is a subject of the study in the field of proof theory.

Although proofs can be written completely in a formal language, for practical reasons, proofs involves a natural language, such as English, and are often expressed as logically organised and clearly worded informal arguments intended to demonstrate that a formal symbolic proof can be constructed. Such arguments are typically easier to check than purely symbolic ones — indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be considered as sufficient to prove a theorem.

Formal and Informal Proofs

In general, a proof is a demonstration that a specified statement follows from a set of assumed statements. The specified statement that follows from the assumed statements is called the conclusion of the proof and the assumed statements that the conclusion follows from are called the premises of the proof.

In mathematics, proofs are often expressed in natural language with some mathematical symbols. This type of proofs are called informal proof. A proof in mathematics is thus an argument showing that the conclusion is a necessary consequence of the premises, i.e. the conclusion must be true if all the premises are true. When all the premises of proofs are statements that have been previously agreed on for the purpose of the study in a given mathematical field, which are called axioms, the conclusions of such proofs are called theorems.

On the other hand, in logic, proofs are formally meant to be sequences of

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 establish that the sum of two even integers is always even:

For any two even integers $x$ and $y$ we can write $x=2a$ and $y=2b$ for some integers $a$ and $b$, since both $x$ and $y$ are multiples of 2. But the sum $x+y=2a+2b=2(a+b)$ is also a multiple of 2, so it is therefore even by definition.

This proof uses definition of even integers, as well as distribution law.

Proof by induction

In proof by induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases. 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.

The principle of mathematical induction states that: 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, ie, P(n) is true for n = 1 (ii) P(m + 1) is true whenever P(m) is true, ie, P(m) is true implies that P(m + 1) is true. Then P(n) is true for the set of natural numbers N.

Proof by transposition

Proof by Transposition establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p".

Proof by contradiction

In proof by contradiction (also known as reductio ad absurdum, Latin for "reduction into the absurd"), it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true. This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that ${\sqrt {2}}$ is irrational:

Suppose that ${\sqrt {2}}$ is rational, so ${\sqrt {2}}={a \over b}$ where a and b are non-zero integers with no common factor (definition of rational number). 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 irrational.

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.

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 colour 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 colour theorem today still has over 600 cases.

Probabilistic proof

A probabilistic proof is one in which an example is shown to exist by methods of probability theory - not an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument'; in the case of the Collatz conjecture it is clear how far that is from a genuine proof. Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.

Combinatorial proof

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Usually a bijection is used to show that the two interpretations give the same result.

Nonconstructive proof

A nonconstructive proof establishes that a certain mathematical object must exist (e.g. "Some X satisfies f(X)"), 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.

Proof nor disproof

There is a class of mathematical formulae for which neither a proof nor disproof exists, using only the standard ZFC axioms. This result is known as Gödel's (first) incompleteness theorem and examples include the continuum hypothesis. Whether a particular unproven proposition can be proved using a standard set of axioms is not always obvious, and can be extremely technical to determine.

Elementary proof

An elementary proof is (usually) a proof which does not use complex analysis. For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.

End of a proof

Sometimes, the abbreviation "Q.E.D." is written to indicate the end of a proof. This abbreviation stands for "Quod Erat Demonstrandum", which is Latin for "that which was to be demonstrated". An alternative is to use a small rectangle with its shorter side horizontal (∎), known as a tombstone or halmos.

See also

• proof theory
• model theory
• computer-aided proof
• automated theorem proving
• invalid proof
• nonconstructive proof
• list of mathematical proofs
• Proofs from THE BOOK

References

ISBN links support NWE through referral fees

• Solow, D. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. Wiley, 2004. ISBN 0-471-68058-3
• Velleman, D. How to Prove It: A Structured Approach. Cambridge University Press, 2006. ISBN 0-521-67599-5

External links

• What are mathematical proofs and why they are important?
• How To Write Proofs by Larry W. Cusick
• How to Write a Proof by Leslie Lamport, and the motivation of proposing such a hierarchical proof style.
• Proofs in Mathematics: Simple, Charming and Fallacious
• The Seventeen Provers of the World, ed. by Freek Wiedijk, foreword by Dana S. Scott, Lecture Notes in Computer Science 3600, Springer, 2006, ISBN 3-540-30704-4. Contains formalized versions of the proof that ${\sqrt {2}}$ is irrational in several automated proof systems.
• What is Proof? Thoughts on proofs and proving.

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