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48-435: Relation or relations may refer to: A finitary or n -ary relation is a set of n -tuples. Specific types of relations include: Relation may also refer to: Finitary relation The non-negative integer n that gives the number of "places" in the relation is called the arity , adicity or degree of the relation. A relation with n "places" is variously called an n -ary relation , an n -adic relation or

96-574: A Boolean domain B be a two-element set, say, B = {0, 1} , whose elements can be interpreted as logical values, typically 0 = false and 1 = true . The characteristic function of R , denoted by χ R , is the Boolean-valued function χ R : X 1 × ... × X n → B , defined by χ R ( ( x 1 , ..., x n ) ) = 1 if Rx 1 ⋯ x n and χ R ( ( x 1 , ..., x n ) ) = 0 otherwise. In applied mathematics, computer science and statistics, it

144-408: A relation of degree n . Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences . When two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some connexion, that connexion is called a relation. Since

192-599: A relational model for databases , thus anticipating the development of data base management systems . Mathematical induction Mathematical induction is a method for proving that a statement P ( n ) {\displaystyle P(n)} is true for every natural number n {\displaystyle n} , that is, that the infinitely many cases P ( 0 ) , P ( 1 ) , P ( 2 ) , P ( 3 ) , … {\displaystyle P(0),P(1),P(2),P(3),\dots }   all hold. This

240-503: A combination of such coins. Let S ( k ) denote the statement " k dollars can be formed by a combination of 4- and 5-dollar coins". The proof that S ( k ) is true for all k ≥ 12 can then be achieved by induction on k as follows: Base case: Showing that S ( k ) holds for k = 12 is simple: take three 4-dollar coins. Induction step: Given that S ( k ) holds for some value of k ≥ 12 ( induction hypothesis ), prove that S ( k + 1) holds, too. Assume S ( k )

288-415: A finite chain of deductive reasoning involving the variable n {\displaystyle n} , which can take infinitely many values. The result is a rigorous proof of the statement, not an assertion of its probability. In 370 BC, Plato 's Parmenides may have contained traces of an early example of an implicit inductive proof, however, the earliest implicit proof by mathematical induction

336-420: A general relation is, and what it is consisted of. For example, databases are designed to deal with empirical data, which is by definition finite, whereas in mathematics, relations with infinite arity (i.e., infinitary relation) are also considered. The logician Augustus De Morgan , in work published around 1860, was the first to articulate the notion of relation in anything like its present sense. He also stated

384-587: A more general version, | sin ⁡ n x | ≤ n | sin ⁡ x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} for any real numbers n , x {\displaystyle n,x} , could be proven without induction; but the case n = 1 2 , x = π {\textstyle n={\frac {1}{2}},\,x=\pi } shows it may be false for non-integer values of n {\displaystyle n} . This suggests we examine

432-501: A statement of the form " x thinks that y likes z ". For instance, the first row states that "Alice thinks that Bob likes Denise". All rows are distinct. The ordering of rows is insignificant but the ordering of columns is significant. The above table is also a simple example of a relational database , a field with theory rooted in relational algebra and applications in data management. Computer scientists, logicians, and mathematicians, however, tend to have different conceptions what

480-408: Is actually a special case of the previous form, because if the statement to be proved is P ( n ) then proving it with these two rules is equivalent with proving P ( n + b ) for all natural numbers n with an induction base case 0 . Assume an infinite supply of 4- and 5-dollar coins. Induction can be used to prove that any whole amount of dollars greater than or equal to 12 can be formed by

528-415: Is an inference rule used in formal proofs , and is the foundation of most correctness proofs for computer programs. Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy , in which the examination of many cases results in a probable conclusion. The mathematical method examines infinitely many cases to prove a general statement, but it does so by

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576-433: Is common to refer to a Boolean-valued function as an n -ary predicate . From the more abstract viewpoint of formal logic and model theory , the relation R constitutes a logical model or a relational structure , that serves as one of many possible interpretations of some n -ary predicate symbol. Because relations arise in many scientific disciplines, as well as in many branches of mathematics and logic , there

624-422: Is considerable variation in terminology. Aside from the set-theoretic extension of a relational concept or term, the term "relation" can also be used to refer to the corresponding logical entity, either the logical comprehension , which is the totality of intensions or abstract properties shared by all elements in the relation, or else the symbols denoting these elements and intensions. Further, some writers of

672-431: Is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis ) and that from each rung we can climb up to

720-497: Is the earliest extant proof of the sum formula for integral cubes . In India, early implicit proofs by mathematical induction appear in Bhaskara 's " cyclic method ". None of these ancient mathematicians, however, explicitly stated the induction hypothesis. Another similar case (contrary to what Vacca has written, as Freudenthal carefully showed) was that of Francesco Maurolico in his Arithmeticorum libri duo (1575), who used

768-409: Is true for some arbitrary k ≥ 12 . If there is a solution for k dollars that includes at least one 4-dollar coin, replace it by a 5-dollar coin to make k + 1 dollars. Otherwise, if only 5-dollar coins are used, k must be a multiple of 5 and so at least 15; but then we can replace three 5-dollar coins by four 4-dollar coins to make k + 1 dollars. In each case, S ( k + 1)

816-414: Is true for some natural number n , it also holds for some strictly smaller natural number m . Because there are no infinite decreasing sequences of natural numbers, this situation would be impossible, thereby showing ( by contradiction ) that Q ( n ) cannot be true for any n . The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on

864-421: Is true. Therefore, by the principle of induction, S ( k ) holds for all k ≥ 12 , and the proof is complete. In this example, although S ( k ) also holds for k ∈ { 4 , 5 , 8 , 9 , 10 } {\textstyle k\in \{4,5,8,9,10\}} , the above proof cannot be modified to replace the minimum amount of 12 dollar to any lower value m . For m = 11 ,

912-1832: Is true. Using the angle addition formula and the triangle inequality , we deduce: | sin ⁡ ( k + 1 ) x | = | sin ⁡ k x cos ⁡ x + sin ⁡ x cos ⁡ k x | (angle addition) ≤ | sin ⁡ k x cos ⁡ x | + | sin ⁡ x cos ⁡ k x | (triangle inequality) = | sin ⁡ k x | | cos ⁡ x | + | sin ⁡ x | | cos ⁡ k x | ≤ | sin ⁡ k x | + | sin ⁡ x | ( | cos ⁡ t | ≤ 1 ) ≤ k | sin ⁡ x | + | sin ⁡ x | (induction hypothesis ) = ( k + 1 ) | sin ⁡ x | . {\displaystyle {\begin{aligned}\left|\sin(k+1)x\right|&=\left|\sin kx\cos x+\sin x\cos kx\right|&&{\text{(angle addition)}}\\&\leq \left|\sin kx\cos x\right|+\left|\sin x\,\cos kx\right|&&{\text{(triangle inequality)}}\\&=\left|\sin kx\right|\left|\cos x\right|+\left|\sin x\right|\left|\cos kx\right|\\&\leq \left|\sin kx\right|+\left|\sin x\right|&&(\left|\cos t\right|\leq 1)\\&\leq k\left|\sin x\right|+\left|\sin x\right|&&{\text{(induction hypothesis}})\\&=(k+1)\left|\sin x\right|.\end{aligned}}} The inequality between

960-409: The binary functions , which relate two inputs and the output. All three of the domains of a homogeneous ternary relation are the same set. Consider the ternary relation R " x thinks that y likes z " over the set of people P = { Alice, Bob, Charles, Denise } , defined by: R can be represented equivalently by the following table: Here, each row represents a triple of R , that is it makes

1008-419: The implication P ( k ) ⟹ P ( k + 1 ) {\displaystyle P(k)\implies P(k+1)} for any natural number k {\displaystyle k} . Assume the induction hypothesis: for a given value n = k ≥ 0 {\displaystyle n=k\geq 0} , the single case P ( k ) {\displaystyle P(k)}

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1056-412: The proof of commutativity accompanying addition of natural numbers . More complicated arguments involving three or more counters are also possible. The method of infinite descent is a variation of mathematical induction which was used by Pierre de Fermat . It is used to show that some statement Q ( n ) is false for all natural numbers n . Its traditional form consists of showing that if Q ( n )

1104-656: The (singleton) set of all 0-tuples. They are sometimes useful for constructing the base case of an induction argument. Unary (1-ary) relations can be viewed as a collection of members (such as the collection of Nobel laureates ) having some property (such as that of having been awarded the Nobel prize ). Every nullary function is a unary relation. Binary (2-ary) relations are the most commonly studied form of finitary relations. Homogeneous binary relations (where X 1 = X 2 ) include Heterogeneous binary relations include Ternary (3-ary) relations include, for example,

1152-456: The base case is actually false; for m = 10 , the second case in the induction step (replacing three 5- by four 4-dollar coins) will not work; let alone for even lower m . It is sometimes desirable to prove a statement involving two natural numbers, n and m , by iterating the induction process. That is, one proves a base case and an induction step for n , and in each of those proves a base case and an induction step for m . See, for example,

1200-458: The definition is predicated on the underlying sets X 1 , ..., X n , R may be more formally defined as the ( n + 1 )-tuple ( X 1 , ..., X n , G ) , where G , called the graph of R , is a subset of the Cartesian product X 1 × ... × X n . As is often done in mathematics, the same symbol is used to refer to the mathematical object and an underlying set, so

1248-554: The exact nature of the property to be proven. All variants of induction are special cases of transfinite induction ; see below . If one wishes to prove a statement, not for all natural numbers, but only for all numbers n greater than or equal to a certain number b , then the proof by induction consists of the following: This can be used, for example, to show that 2 ≥ n + 5 for n ≥ 3 . In this way, one can prove that some statement P ( n ) holds for all n ≥ 1 , or even for all n ≥ −5 . This form of mathematical induction

1296-402: The extreme left hand and right hand sides, we deduce that: 0 + 1 + 2 + ⋯ + k + ( k + 1 ) = ( k + 1 ) ( ( k + 1 ) + 1 ) 2 . {\displaystyle 0+1+2+\cdots +k+(k+1)={\frac {(k+1)((k+1)+1)}{2}}.} That is, the statement P ( k + 1) also holds true, establishing

1344-454: The extreme left-hand and right-hand quantities shows that P ( k + 1 ) {\displaystyle P(k+1)} is true, which completes the induction step. Conclusion: The proposition P ( n ) {\displaystyle P(n)} holds for all natural numbers n . {\displaystyle n.}   Q.E.D. In practice, proofs by induction are often structured differently, depending on

1392-490: The first formal results in the theory of relations (on De Morgan and relations, see Merrill 1990). Charles Peirce , Gottlob Frege , Georg Cantor , Richard Dedekind and others advanced the theory of relations. Many of their ideas, especially on relations called orders , were summarized in The Principles of Mathematics (1903) where Bertrand Russell made free use of these results. In 1970, Edgar Codd proposed

1440-1238: The following statement P ( n ) for all natural numbers n . P ( n ) :     0 + 1 + 2 + ⋯ + n = n ( n + 1 ) 2 . {\displaystyle P(n)\!:\ \ 0+1+2+\cdots +n={\frac {n(n+1)}{2}}.} This states a general formula for the sum of the natural numbers less than or equal to a given number; in fact an infinite sequence of statements: 0 = ( 0 ) ( 0 + 1 ) 2 {\displaystyle 0={\tfrac {(0)(0+1)}{2}}} , 0 + 1 = ( 1 ) ( 1 + 1 ) 2 {\displaystyle 0+1={\tfrac {(1)(1+1)}{2}}} , 0 + 1 + 2 = ( 2 ) ( 2 + 1 ) 2 {\displaystyle 0+1+2={\tfrac {(2)(2+1)}{2}}} , etc. Proposition. For every n ∈ N {\displaystyle n\in \mathbb {N} } , 0 + 1 + 2 + ⋯ + n = n ( n + 1 ) 2 . {\displaystyle 0+1+2+\cdots +n={\tfrac {n(n+1)}{2}}.} Proof. Let P ( n ) be

1488-592: The induction hypothesis that for a particular k , the single case n = k holds, meaning P ( k ) is true: 0 + 1 + ⋯ + k = k ( k + 1 ) 2 . {\displaystyle 0+1+\cdots +k={\frac {k(k+1)}{2}}.} It follows that: ( 0 + 1 + 2 + ⋯ + k ) + ( k + 1 ) = k ( k + 1 ) 2 + ( k + 1 ) . {\displaystyle (0+1+2+\cdots +k)+(k+1)={\frac {k(k+1)}{2}}+(k+1).} Algebraically ,

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1536-492: The induction step, that the statement holds for a particular n , is called the induction hypothesis or inductive hypothesis . To prove the induction step, one assumes the induction hypothesis for n and then uses this assumption to prove that the statement holds for n + 1 . Authors who prefer to define natural numbers to begin at 0 use that value in the base case; those who define natural numbers to begin at 1 use that value. Mathematical induction can be used to prove

1584-651: The induction step. Conclusion: Since both the base case and the induction step have been proved as true, by mathematical induction the statement P ( n ) holds for every natural number n . Q.E.D. Induction is often used to prove inequalities . As an example, we prove that | sin ⁡ n x | ≤ n | sin ⁡ x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} for any real number x {\displaystyle x} and natural number n {\displaystyle n} . At first glance, it may appear that

1632-420: The latter persuasion introduce terms with more concrete connotations (such as "relational structure" for the set-theoretic extension of a given relational concept). Nullary (0-ary) relations count only two members: the empty nullary relation, which never holds, and the universal nullary relation, which always holds. This is because there is only one 0-tuple, the empty tuple (), and there are exactly two subsets of

1680-501: The next case n = k + 1 {\displaystyle n=k+1} . These two steps establish that the statement holds for every natural number n {\displaystyle n} . The base case does not necessarily begin with n = 0 {\displaystyle n=0} , but often with n = 1 {\displaystyle n=1} , and possibly with any fixed natural number n = N {\displaystyle n=N} , establishing

1728-414: The next one (the step ). A proof by induction consists of two cases. The first, the base case , proves the statement for n = 0 {\displaystyle n=0} without assuming any knowledge of other cases. The second case, the induction step , proves that if the statement holds for any given case n = k {\displaystyle n=k} , then it must also hold for

1776-623: The right hand side simplifies as: k ( k + 1 ) 2 + ( k + 1 ) = k ( k + 1 ) + 2 ( k + 1 ) 2 = ( k + 1 ) ( k + 2 ) 2 = ( k + 1 ) ( ( k + 1 ) + 1 ) 2 . {\displaystyle {\begin{aligned}{\frac {k(k+1)}{2}}+(k+1)&={\frac {k(k+1)+2(k+1)}{2}}\\&={\frac {(k+1)(k+2)}{2}}\\&={\frac {(k+1)((k+1)+1)}{2}}.\end{aligned}}} Equating

1824-619: The statement | sin ⁡ n x | ≤ n | sin ⁡ x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} . We induce on n {\displaystyle n} . Base case: The calculation | sin ⁡ 0 x | = 0 ≤ 0 = 0 | sin ⁡ x | {\displaystyle \left|\sin 0x\right|=0\leq 0=0\left|\sin x\right|} verifies P ( 0 ) {\displaystyle P(0)} . Induction step: We show

1872-609: The statement 0 + 1 + 2 + ⋯ + n = n ( n + 1 ) 2 . {\displaystyle 0+1+2+\cdots +n={\tfrac {n(n+1)}{2}}.} We give a proof by induction on n . Base case: Show that the statement holds for the smallest natural number n = 0 . P (0) is clearly true: 0 = 0 ( 0 + 1 ) 2 . {\displaystyle 0={\tfrac {0(0+1)}{2}}\,.} Induction step: Show that for every k ≥ 0 , if P ( k ) holds, then P ( k + 1) also holds. Assume

1920-467: The statement P ( n ) defined as " Q ( m ) is false for all natural numbers m less than or equal to n ", it follows that P ( n ) holds for all n , which means that Q ( n ) is false for every natural number n . If one wishes to prove that a property P holds for all natural numbers less than or equal to n , proving P satisfies the following conditions suffices: The most common form of proof by mathematical induction requires proving in

1968-452: The statement ( x 1 , ..., x n ) ∈ R is often used to mean ( x 1 , ..., x n ) ∈ G is read " x 1 , ..., x n are R -related" and are denoted using prefix notation by Rx 1 ⋯ x n and using postfix notation by x 1 ⋯ x n R . In the case where R is a binary relation, those statements are also denoted using infix notation by x 1 Rx 2 . The following considerations apply: Let

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2016-690: The statement specifically for natural values of n {\displaystyle n} , and induction is the readiest tool. Proposition. For any x ∈ R {\displaystyle x\in \mathbb {R} } and n ∈ N {\displaystyle n\in \mathbb {N} } , | sin ⁡ n x | ≤ n | sin ⁡ x | {\displaystyle \left|\sin nx\right|\leq n\left|\sin x\right|} . Proof. Fix an arbitrary real number x {\displaystyle x} , and let P ( n ) {\displaystyle P(n)} be

2064-423: The technique to prove that the sum of the first n odd integers is n . The earliest rigorous use of induction was by Gersonides (1288–1344). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665). Another Frenchman, Fermat , made ample use of a related principle: indirect proof by infinite descent . The induction hypothesis

2112-447: The truth of the statement for all natural numbers n ≥ N {\displaystyle n\geq N} . The method can be extended to prove statements about more general well-founded structures, such as trees ; this generalization, known as structural induction , is used in mathematical logic and computer science . Mathematical induction in this extended sense is closely related to recursion . Mathematical induction

2160-419: The two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 1 ) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri

2208-590: Was also employed by the Swiss Jakob Bernoulli , and from then on it became well known. The modern formal treatment of the principle came only in the 19th century, with George Boole , Augustus De Morgan , Charles Sanders Peirce , Giuseppe Peano , and Richard Dedekind . The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n . The proof consists of two steps: The hypothesis in

2256-462: Was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n . He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence

2304-516: Was written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle . Whilst the original work was lost, it was later referenced by Al-Samawal al-Maghribi in his treatise al-Bahir fi'l-jabr (The Brilliant in Algebra) in around 1150 AD. Katz says in his history of mathematics Another important idea introduced by al-Karaji and continued by al-Samaw'al and others

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