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86-661: It is likely that the FX-501P/502P were the first LCD programmable calculators to be produced as up until 1979 (and the introduction of the HP-41C) no manufacturer had introduced such a device. The FX-502P series use algebraic logic as was state-of-the-art at the time. The FX-501P and FX-502P featured a single line 7-segment liquid crystal display with 10 digits as main display. An additional 3 digits 7-segment display used to display exponents and program steps when entering or debugging programs and 10 status indicators. The display

172-428: A , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table. In mathematics , a binary relation associates elements of one set called

258-653: A complete lattice . In some systems of axiomatic set theory , relations are extended to classes , which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox . A binary relation is the most studied special case n = 2 {\displaystyle n=2} of an n {\displaystyle n} -ary relation over sets X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} , which

344-526: A heterogeneous relation . The prefix hetero is from the Greek ἕτερος ( heteros , "other, another, different"). A heterogeneous relation has been called a rectangular relation , suggesting that it does not have the square-like symmetry of a homogeneous relation on a set where A = B . {\displaystyle A=B.} Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "... a variant of

430-508: A least upper bound (also called supremum) in R . {\displaystyle \mathbb {R} .} However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ {\displaystyle \leq } to the rational numbers. A binary relation R {\displaystyle R} over sets X {\displaystyle X} and Y {\displaystyle Y}

516-401: A semigroup with involution . Some important properties that a homogeneous relation R {\displaystyle R} over a set X {\displaystyle X} may have are: A partial order is a relation that is reflexive, antisymmetric, and transitive. A strict partial order is a relation that is irreflexive, asymmetric, and transitive. A total order

602-463: A binary relation is the " divides " relation over the set of prime numbers P {\displaystyle \mathbb {P} } and the set of integers Z {\displaystyle \mathbb {Z} } , in which each prime p {\displaystyle p} is related to each integer z {\displaystyle z} that is a multiple of p {\displaystyle p} , but not to an integer that

688-575: A binary relation over every set and its power set. A homogeneous relation over a set X {\displaystyle X} is a binary relation over X {\displaystyle X} and itself, i.e. it is a subset of the Cartesian product X × X . {\displaystyle X\times X.} It is also simply called a (binary) relation over X {\displaystyle X} . A homogeneous relation R {\displaystyle R} over

774-464: A logician stems mainly from the work of Wolfgang Lenzen, summarized in Lenzen (2004) . To see how present-day work in logic and metaphysics can draw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000) . Historical perspective Binary relation All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all

860-434: A relation is reflexive , irreflexive, symmetric , antisymmetric , asymmetric , transitive , total , trichotomous , a partial order , total order , strict weak order , total preorder (weak order), or an equivalence relation , then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting

946-434: A set X {\displaystyle X} is the power set 2 X × X {\displaystyle 2^{X\times X}} which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation . Considering composition of relations as a binary operation on B ( X ) {\displaystyle {\mathcal {B}}(X)} , it forms

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1032-592: A set X {\displaystyle X} may be identified with a directed simple graph permitting loops , where X {\displaystyle X} is the vertex set and R {\displaystyle R} is the edge set (there is an edge from a vertex x {\displaystyle x} to a vertex y {\displaystyle y} if and only if x R y {\displaystyle xRy} ). The set of all homogeneous relations B ( X ) {\displaystyle {\mathcal {B}}(X)} over

1118-492: A specific set A {\displaystyle A} ): the resulting set relation can be denoted by ⊆ A . {\displaystyle \subseteq _{A}.} Also, the "member of" relation needs to be restricted to have domain A {\displaystyle A} and codomain P ( A ) {\displaystyle P(A)} to obtain a binary relation ∈ A {\displaystyle \in _{A}} that

1204-447: A theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple ( X , Y , G ) {\displaystyle (X,Y,G)} , as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.) With this definition one can for instance define

1290-404: A total order on natural numbers N , {\displaystyle \mathbb {N} ,} " x < y {\displaystyle x<y} " is a strict total order on N , {\displaystyle \mathbb {N} ,} and " x {\displaystyle x} is parallel to y {\displaystyle y} " is an equivalence relation on

1376-489: A univalent total relation is a function . The formula for totality is I ⊆ R R T . {\displaystyle I\subseteq RR^{T}.} Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation. The facility of complementary relations inspired Augustus De Morgan and Ernst Schröder to introduce equivalences using R ¯ {\displaystyle {\bar {R}}} for

1462-451: A wide variety of concepts. These include, among others: A function may be defined as a binary relation that meets additional constraints. Binary relations are also heavily used in computer science . A binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} is an element of the power set of X × Y . {\displaystyle X\times Y.} Since

1548-555: Is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother. Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation ≤ {\displaystyle \leq } is that every non-empty subset S ⊆ R {\displaystyle S\subseteq \mathbb {R} } with an upper bound in R {\displaystyle \mathbb {R} } has

1634-785: Is a binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} , and S {\displaystyle S} is a binary relation over sets Y {\displaystyle Y} and Z {\displaystyle Z} then S ∘ R = { ( x , z ) ∣  there exists  y ∈ Y  such that  x R y  and  y S z } {\displaystyle S\circ R=\{(x,z)\mid {\text{ there exists }}y\in Y{\text{ such that }}xRy{\text{ and }}ySz\}} (also denoted by R ; S {\displaystyle R;S} )

1720-401: Is a relation that is reflexive, antisymmetric, transitive and connected. A strict total order is a relation that is irreflexive, asymmetric, transitive and connected. An equivalence relation is a relation that is reflexive, symmetric, and transitive. For example, " x {\displaystyle x} divides y {\displaystyle y} " is a partial, but not

1806-420: Is a set. Bertrand Russell has shown that assuming ∈ {\displaystyle \in } to be defined over all sets leads to a contradiction in naive set theory , see Russell's paradox . Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory , and allow the domain and codomain (and so the graph) to be proper classes : in such

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1892-585: Is a subset of X {\displaystyle X} then R | S = { ( x , y ) ∣ x R y  and  x ∈ S } {\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S\}} is the left-restriction relation of R {\displaystyle R} to S {\displaystyle S} over X {\displaystyle X} and Y {\displaystyle Y} . If

1978-419: Is a subset of X × Y . {\displaystyle X\times Y.} The set X {\displaystyle X} is called the domain or set of departure of R {\displaystyle R} , and the set Y {\displaystyle Y} the codomain or set of destination of R {\displaystyle R} . In order to specify

2064-983: Is a subset of the Cartesian product X 1 × ⋯ × X n . {\displaystyle X_{1}\times \cdots \times X_{n}.} Given sets X {\displaystyle X} and Y {\displaystyle Y} , the Cartesian product X × Y {\displaystyle X\times Y} is defined as { ( x , y ) ∣ x ∈ X  and  y ∈ Y } , {\displaystyle \{(x,y)\mid x\in X{\text{ and }}y\in Y\},} and its elements are called ordered pairs . A binary relation R {\displaystyle R} over sets X {\displaystyle X} and Y {\displaystyle Y}

2150-404: Is contained in S {\displaystyle S} and S {\displaystyle S} is contained in R {\displaystyle R} , then R {\displaystyle R} and S {\displaystyle S} are called equal written R = S {\displaystyle R=S} . If R {\displaystyle R}

2236-417: Is contained in S {\displaystyle S} but S {\displaystyle S} is not contained in R {\displaystyle R} , then R {\displaystyle R} is said to be smaller than S {\displaystyle S} , written R ⊊ S . {\displaystyle R\subsetneq S.} For example, on

2322-507: Is equal to its converse if and only if it is symmetric . If R {\displaystyle R} is a binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} then R ¯ = { ( x , y ) ∣ ¬ x R y } {\displaystyle {\bar {R}}=\{(x,y)\mid \neg xRy\}} (also denoted by ¬ R {\displaystyle \neg R} )

2408-447: Is found in the power set of X × Y , where X ≠ Y . Whether a given relation holds for two individuals is one bit of information, so relations are studied with Boolean arithmetic. Elements of the power set are partially ordered by inclusion , and lattice of these sets becomes an algebra through relative multiplication or composition of relations . "The basic operations are set-theoretic union, intersection and complementation,

2494-409: Is in Y {\displaystyle Y} . It encodes the common concept of relation: an element x {\displaystyle x} is related to an element y {\displaystyle y} , if and only if the pair ( x , y ) {\displaystyle (x,y)} belongs to the set of ordered pairs that defines the binary relation. An example of

2580-513: Is not a multiple of p {\displaystyle p} . In this relation, for instance, the prime number 2 {\displaystyle 2} is related to numbers such as − 4 {\displaystyle -4} , 0 {\displaystyle 0} , 6 {\displaystyle 6} , 10 {\displaystyle 10} , but not to 1 {\displaystyle 1} or 9 {\displaystyle 9} , just as

2666-623: Is not necessary that X = Y {\displaystyle X=Y} . Since relations are sets, they can be manipulated using set operations, including union , intersection , and complementation , and satisfying the laws of an algebra of sets . Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations , for which there are textbooks by Ernst Schröder , Clarence Lewis , and Gunther Schmidt . A deeper analysis of relations involves decomposing them into subsets called concepts , and placing them in

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2752-431: Is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems ) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under

2838-499: Is represented as a member of the power set of A × B with properties described by Boolean algebra . The "calculus of relations" is arguably the culmination of Leibniz's approach to logic. At the Hochschule Karlsruhe the calculus of relations was described by Ernst Schröder . In particular he formulated Schröder rules , though De Morgan had anticipated them with his Theorem K. In 1903 Bertrand Russell developed

2924-772: Is said to be contained in a relation S {\displaystyle S} over X {\displaystyle X} and Y {\displaystyle Y} , written R ⊆ S , {\displaystyle R\subseteq S,} if R {\displaystyle R} is a subset of S {\displaystyle S} , that is, for all x ∈ X {\displaystyle x\in X} and y ∈ Y , {\displaystyle y\in Y,} if x R y {\displaystyle xRy} , then x S y {\displaystyle xSy} . If R {\displaystyle R}

3010-1018: Is the complementary relation of R {\displaystyle R} over X {\displaystyle X} and Y {\displaystyle Y} . For example, = {\displaystyle =} and ≠ {\displaystyle \neq } are each other's complement, as are ⊆ {\displaystyle \subseteq } and ⊈ {\displaystyle \not \subseteq } , ⊇ {\displaystyle \supseteq } and ⊉ {\displaystyle \not \supseteq } , ∈ {\displaystyle \in } and ∉ {\displaystyle \not \in } , and for total orders also < {\displaystyle <} and ≥ {\displaystyle \geq } , and > {\displaystyle >} and ≤ {\displaystyle \leq } . The complement of

3096-505: Is the composition relation of R {\displaystyle R} and S {\displaystyle S} over X {\displaystyle X} and Z {\displaystyle Z} . The identity element is the identity relation. The order of R {\displaystyle R} and S {\displaystyle S} in the notation S ∘ R , {\displaystyle S\circ R,} used here agrees with

3182-654: Is the converse relation , also called inverse relation , of R {\displaystyle R} over Y {\displaystyle Y} and X {\displaystyle X} . For example, = {\displaystyle =} is the converse of itself, as is ≠ {\displaystyle \neq } , and < {\displaystyle <} and > {\displaystyle >} are each other's converse, as are ≤ {\displaystyle \leq } and ≥ {\displaystyle \geq } . A binary relation

3268-454: Is the intersection relation of R {\displaystyle R} and S {\displaystyle S} over X {\displaystyle X} and Y {\displaystyle Y} . The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2". If R {\displaystyle R}

3354-406: Is the restriction relation of R {\displaystyle R} to S {\displaystyle S} over X {\displaystyle X} . If R {\displaystyle R} is a binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} and if S {\displaystyle S}

3440-425: Is the union relation of R {\displaystyle R} and S {\displaystyle S} over X {\displaystyle X} and Y {\displaystyle Y} . The identity element is the empty relation. For example, ≤ {\displaystyle \leq } is the union of < and =, and ≥ {\displaystyle \geq }

3526-401: Is the identity relation on the range of R . The injective property corresponds to univalence of R T {\displaystyle R^{T}} , or the formula R R T ⊆ I , {\displaystyle RR^{T}\subseteq I,} where this time I is the identity on the domain of R . But a univalent relation is only a partial function , while

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3612-643: Is the parent of y {\displaystyle y} and y {\displaystyle y} is the mother of z {\displaystyle z} , then x {\displaystyle x} is the maternal grandparent of z {\displaystyle z} . If R {\displaystyle R} is a binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} then R T = { ( y , x ) ∣ x R y } {\displaystyle R^{\textsf {T}}=\{(y,x)\mid xRy\}}

3698-570: Is the set of all x {\displaystyle x} such that x R y {\displaystyle xRy} for at least one y {\displaystyle y} . The codomain of definition , active codomain , image or range of R {\displaystyle R} is the set of all y {\displaystyle y} such that x R y {\displaystyle xRy} for at least one x {\displaystyle x} . The field of R {\displaystyle R}

3784-463: Is the union of > and =. If R {\displaystyle R} and S {\displaystyle S} are binary relations over sets X {\displaystyle X} and Y {\displaystyle Y} then R ∩ S = { ( x , y ) ∣ x R y  and  x S y } {\displaystyle R\cap S=\{(x,y)\mid xRy{\text{ and }}xSy\}}

3870-411: Is the union of its domain of definition and its codomain of definition. When X = Y , {\displaystyle X=Y,} a binary relation is called a homogeneous relation (or endorelation ). To emphasize the fact that X {\displaystyle X} and Y {\displaystyle Y} are allowed to be different, a binary relation is also called

3956-506: Is to select a "large enough" set A {\displaystyle A} , that contains all the objects of interest, and work with the restriction = A {\displaystyle =_{A}} instead of = {\displaystyle =} . Similarly, the "subset of" relation ⊆ {\displaystyle \subseteq } needs to be restricted to have domain and codomain P ( A ) {\displaystyle P(A)} (the power set of

4042-524: The graph of the binary relation. The statement ( x , y ) ∈ R {\displaystyle (x,y)\in R} reads " x {\displaystyle x} is R {\displaystyle R} -related to y {\displaystyle y} " and is denoted by x R y {\displaystyle xRy} . The domain of definition or active domain of R {\displaystyle R}

4128-488: The Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over X {\displaystyle X} and Y {\displaystyle Y} and a relation over Y {\displaystyle Y} and Z {\displaystyle Z} ),

4214-518: The FA-1 to store program and data to Compact Cassette using the Kansas City standard . The FA-1 also enabled the calculators to generate musical notes. The FX-501P was used on the 1981 song "Pocket Calculator" by electronic music group Kraftwerk . Algebraic logic In mathematical logic , algebraic logic is the reasoning obtained by manipulating equations with free variables . What

4300-537: The Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations (when X = Y {\displaystyle X=Y} ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation. While

4386-391: The converse relation R T {\displaystyle R^{\textsf {T}}} is the converse of the complement: R T ¯ = R ¯ T . {\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.} If X = Y , {\displaystyle X=Y,}

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4472-444: The domain with elements of another set called the codomain . Precisely, a binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} is a set of ordered pairs ( x , y ) {\displaystyle (x,y)} where x {\displaystyle x} is in X {\displaystyle X} and y {\displaystyle y}

4558-579: The rational numbers , the relation > {\displaystyle >} is smaller than ≥ {\displaystyle \geq } , and equal to the composition > ∘ > {\displaystyle >\circ >} . Binary relations over sets X {\displaystyle X} and Y {\displaystyle Y} can be represented algebraically by logical matrices indexed by X {\displaystyle X} and Y {\displaystyle Y} with entries in

4644-456: The transpose of a staircase. Riguet generated rectangular relations by taking the outer product of logical vectors; these contribute to the non-enlargeable rectangles of formal concept analysis . Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before the Parkinson and Loemker translations. Our present understanding of Leibniz as

4730-531: The 1910–13 publication of Principia Mathematica , and Tarski revived interest in relations with his 1941 essay "On the Calculus of Relations". According to Helena Rasiowa , "The years 1920-40 saw, in particular in the Polish school of logic, researches on non-classical propositional calculi conducted by what is termed the logical matrix method. Since logical matrices are certain abstract algebras, this led to

4816-536: The 2nd example relation is surjective (see below ), the 1st is not. Some important types of binary relations R {\displaystyle R} over sets X {\displaystyle X} and Y {\displaystyle Y} are listed below. Uniqueness properties: Totality properties (only definable if the domain X {\displaystyle X} and codomain Y {\displaystyle Y} are specified): Uniqueness and totality properties (only definable if

4902-474: The FX-501 ;/ FX-502P from its competitors was that programming was retained in a battery-buffered memory when the calculator was turned off. Here is a sample program that computes the factorial of an integer number from 2 to 69. For 5!, the user would type 5 P0 and get the result 120. The whole program is only 9 bytes long. Compact Cassette  via one of: The FX-501P and FX-502P used

4988-644: The calculus of relations and logicism as his version of pure mathematics based on the operations of the calculus as primitive notions . The "Boole–Schröder algebra of logic" was developed at University of California, Berkeley in a textbook by Clarence Lewis in 1918. He treated the logic of relations as derived from the propositional functions of two or more variables. Hugh MacColl , Gottlob Frege , Giuseppe Peano , and A. N. Whitehead all shared Leibniz's dream of combining symbolic logic , mathematics , and philosophy . Some writings by Leopold Löwenheim and Thoralf Skolem on algebraic logic appeared after

5074-448: The calculus of relations, Jacques Riguet used the algebraic logic to advance useful concepts: he extended the concept of an equivalence relation (on a set) to the heterogeneous case with the notion of a difunctional relation. Riguet also extended ordering to the heterogeneous context by his note that a staircase logical matrix has a complement that is also a staircase, and that the theorem of N. M. Ferrers follows from interpretation of

5160-434: The choices of the sets X {\displaystyle X} and Y {\displaystyle Y} , some authors define a binary relation or correspondence as an ordered triple ( X , Y , G ) {\displaystyle (X,Y,G)} , where G {\displaystyle G} is a subset of X × Y {\displaystyle X\times Y} called

5246-642: The complement has the following properties: If R {\displaystyle R} is a binary homogeneous relation over a set X {\displaystyle X} and S {\displaystyle S} is a subset of X {\displaystyle X} then R | S = { ( x , y ) ∣ x R y  and  x ∈ S  and  y ∈ S } {\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S{\text{ and }}y\in S\}}

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5332-456: The complement of relation R . These equivalences provide alternative formulas for univalent relations ( R I ¯ ⊆ R ¯ {\displaystyle R{\bar {I}}\subseteq {\bar {R}}} ), and total relations ( R ¯ ⊆ R I ¯ {\displaystyle {\bar {R}}\subseteq R{\bar {I}}} ). Therefore, mappings satisfy

5418-461: The domain X {\displaystyle X} and codomain Y {\displaystyle Y} are specified): If relations over proper classes are allowed: Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory . For example, to model

5504-464: The first of several works on the logic of relatives . Alexander Macfarlane published his Principles of the Algebra of Logic in 1879, and in 1883, Christine Ladd , a student of Peirce at Johns Hopkins University , published "On the Algebra of Logic". Logic turned more algebraic when binary relations were combined with composition of relations . For sets A and B , a relation over A and B

5590-454: The formula R ¯ = R I ¯ . {\displaystyle {\bar {R}}=R{\bar {I}}.} Schmidt uses this principle as "slipping below negation from the left". For a mapping f ,  f A ¯ = f A ¯ . {\displaystyle f{\bar {A}}={\overline {fA}}.} The relation algebra structure, based in set theory,

5676-439: The general concept of "equality" as a binary relation = {\displaystyle =} , take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory. In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem

5762-455: The inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules , provides a calculus to work in the power set of A × B . {\displaystyle A\times B.} In contrast to homogeneous relations, the composition of relations operation is only a partial function . The necessity of matching target to source of composed relations has led to

5848-428: The latter set is ordered by inclusion ( ⊆ {\displaystyle \subseteq } ), each relation has a place in the lattice of subsets of X × Y . {\displaystyle X\times Y.} A binary relation is called a homogeneous relation when X = Y {\displaystyle X=Y} . A binary relation is also called a heterogeneous relation when it

5934-699: The oldest approach to formal logic, arguably beginning with a number of memoranda Leibniz wrote in the 1680s, some of which were published in the 19th century and translated into English by Clarence Lewis in 1918. But nearly all of Leibniz's known work on algebraic logic was published only in 1903 after Louis Couturat discovered it in Leibniz's Nachlass . Parkinson (1966) and Loemker (1969) translated selections from Couturat's volume into English. Modern mathematical logic began in 1847, with two pamphlets whose respective authors were George Boole and Augustus De Morgan . In 1870 Charles Sanders Peirce published

6020-440: The prime number 3 {\displaystyle 3} is related to 0 {\displaystyle 0} , 6 {\displaystyle 6} , and 9 {\displaystyle 9} , but not to 4 {\displaystyle 4} or 13 {\displaystyle 13} . Binary relations, and especially homogeneous relations , are used in many branches of mathematics to model

6106-457: The programming model could be considered Turing complete . Since the FX-501P and FX-502P only employed a seven-segment display each program step was represented by a special 2-digit codes made up of the digits 0 .. 9 and the character C, E, F and P. The calculator came with a special overlay so the user did not need to memorize the mapping between code and actual command. What differentiated

6192-406: The relation " x {\displaystyle x} is parent of y {\displaystyle y} " to females yields the relation " x {\displaystyle x} is mother of the woman y {\displaystyle y} "; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of"

6278-494: The relative multiplication, and conversion." The conversion refers to the converse relation that always exists, contrary to function theory. A given relation may be represented by a logical matrix ; then the converse relation is represented by the transpose matrix. A relation obtained as the composition of two others is then represented by the logical matrix obtained by matrix multiplication using Boolean arithmetic. An example of calculus of relations arises in erotetics ,

6364-458: The right question to elicit a sufficient answer is recognized in Socratic method dialogue. The description of the key binary relation properties has been formulated with the calculus of relations. The univalence property of functions describes a relation R that satisfies the formula R T R ⊆ I , {\displaystyle R^{T}R\subseteq I,} where I

6450-516: The set of all lines in the Euclidean plane . All operations defined in section § Operations also apply to homogeneous relations. Beyond that, a homogeneous relation over a set X {\displaystyle X} may be subjected to closure operations like: Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets , extended by composition of relations and

6536-413: The standard notational order for composition of functions . For example, the composition (is parent of) ∘ {\displaystyle \circ } (is mother of) yields (is maternal grandparent of), while the composition (is mother of) ∘ {\displaystyle \circ } (is parent of) yields (is grandmother of). For the former case, if x {\displaystyle x}

6622-502: The table below, the left column contains one or more logical or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators." Algebraic formalisms going beyond first-order logic in at least some respects include: Algebraic logic is, perhaps,

6708-1121: The term "correspondence" for a binary relation with reference to X {\displaystyle X} and Y {\displaystyle Y} . In a binary relation, the order of the elements is important; if x ≠ y {\displaystyle x\neq y} then y R x {\displaystyle yRx} can be true or false independently of x R y {\displaystyle xRy} . For example, 3 {\displaystyle 3} divides 9 {\displaystyle 9} , but 9 {\displaystyle 9} does not divide 3 {\displaystyle 3} . If R {\displaystyle R} and S {\displaystyle S} are binary relations over sets X {\displaystyle X} and Y {\displaystyle Y} then R ∪ S = { ( x , y ) ∣ x R y  or  x S y } {\displaystyle R\cup S=\{(x,y)\mid xRy{\text{ or }}xSy\}}

6794-601: The theory has evolved that treats relations from the very beginning as heterogeneous or rectangular , i.e. as relations where the normal case is that they are relations between different sets." The terms correspondence , dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product X × Y {\displaystyle X\times Y} without reference to X {\displaystyle X} and Y {\displaystyle Y} , and reserve

6880-447: The theory of questions. In the universe of utterances there are statements S and questions Q . There are two relations π and α from Q to S : q α a holds when a is a direct answer to question q . The other relation, q π p holds when p is a presupposition of question q . The converse relation π runs from S to Q so that the composition π α is a homogeneous relation on S . The art of putting

6966-453: The umbrella of classical algebraic logic ( Czelakowski 2003 ). Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator ( Czelakowski 2003 ). A homogeneous binary relation is found in the power set of X × X for some set X , while a heterogeneous relation

7052-593: The use of converse relations . The inclusion R ⊆ S , {\displaystyle R\subseteq S,} meaning that a R b {\displaystyle aRb} implies a S b {\displaystyle aSb} , sets the scene in a lattice of relations. But since P ⊆ Q ≡ ( P ∩ Q ¯ = ∅ ) ≡ ( P ∩ Q = P ) , {\displaystyle P\subseteq Q\equiv (P\cap {\bar {Q}}=\varnothing )\equiv (P\cap Q=P),}

7138-399: The use of an algebraic method in logic." Brady (2000) discusses the rich historical connections between algebraic logic and model theory . The founders of model theory, Ernst Schröder and Leopold Loewenheim, were logicians in the algebraic tradition. Alfred Tarski , the founder of set theoretic model theory as a major branch of contemporary mathematical logic, also: In the practice of

7224-577: Was covered with a yellow filter, supposedly to prevent ultra-violet radiation damage to the unstable (at the time) Liquid Crystal compound. They were the first Casio calculators to implement engineering notation , and the first calculators in general to implement an engineering notation with shift facility (following Hewlett Packard calculators implementing the first engineering notation in general, and some Commodore and Texas Instruments calculators implementing variable scientific notation with exponent shift facility). The programming model employed

7310-716: Was key stroke programming by which each key pressed was recorded and later played back. On record multiple key presses were merged into a single programming step. All operations fitted into one program step. The FX-501P could store 128 steps, with 11 memory registers. The FX-502P had twice that capacity with 256 steps and 22 memory registers. Conditional and unconditional jumps as well as subroutines were supported. The FX-502P series supported 10 labels for programs and subroutines called P0 .. P9. Each program or subroutine could have up to 10 local labels called LBL0 .. LBL9 for jumps and branches. The FX-501P and FX-502P supported indirect addressing both for memory access and jumps and therefore

7396-412: Was transcended by Tarski with axioms describing it. Then he asked if every algebra satisfying the axioms could be represented by a set relation. The negative answer opened the frontier of abstract algebraic logic . Algebraic logic treats algebraic structures , often bounded lattices , as models (interpretations) of certain logics , making logic a branch of order theory . In algebraic logic: In

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