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Transitive Relation
In mathematics, a binary relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If and then ; and if and then . Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie. On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does not follow that Alice is the birth mother ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime p is related to each integer z that is a Divisibility, multiple of p, but not to an integer that is not a Multiple (mathematics), multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Successor Function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(''a'', ''b'') = 1 + ''b''. In this context, the extension of zeration is addition, which is defined as repeated succession. Overview The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers. In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition are defined. For example, 1 is defined to be ''S''(0), and addition on natural numbers is defined recursively by: : This can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreflexive Relation
In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Etymology The word ''reflexive'' is originally derived from the Medieval Latin ''reflexivus'' ('recoiling' reflex.html" ;"title="f. ''reflex">f. ''reflex'' or 'directed upon itself') (c. 1250 AD) from the classical Latin ''reflexus-'' ('turn away', 'reflection') + ''-īvus'' (suffix). The word entered Early Modern English in the 1580s. The sense of the word meaning 'directed upon itself', as now used in mathematics, surviving mostly by its use in philosophy and grammar (cf. ''Reflexive verb'' and ''Reflexive pronoun''). The first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymmetric Relation
In mathematics, an asymmetric relation is a binary relation R on a set X where for all a, b \in X, if a is related to b then b is ''not'' related to a. Formal definition Preliminaries A binary relation on X is any subset R of X \times X. Given a, b \in X, write a R b if and only if (a, b) \in R, which means that a R b is shorthand for (a, b) \in R. The expression a R b is read as "a is related to b by R." Definition The binary relation R is called if for all a, b \in X, if a R b is true then b R a is false; that is, if (a, b) \in R then (b, a) \not\in R. This can be written in the notation of first-order logic as \forall a, b \in X: a R b \implies \lnot(b R a). A logically equivalent definition is: :for all a, b \in X, at least one of a R b and b R a is , which in first-order logic can be written as: \forall a, b \in X: \lnot(a R b \wedge b R a). A relation is asymmetric if and only if it is both antisymmetric and irreflexive, so this may also be taken as a definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franklin Pierce
Franklin Pierce (November 23, 1804October 8, 1869) was the 14th president of the United States, serving from 1853 to 1857. A northern Democratic Party (United States), Democrat who believed that the Abolitionism in the United States, abolitionist movement was a fundamental threat to the nation's unity, he alienated anti-slavery groups by signing the Kansas–Nebraska Act and enforcing the Fugitive Slave Act. Conflict between North and South continued after Pierce's presidency, and, after Abraham Lincoln was 1860 United States presidential election, elected president in 1860, the Confederate States of America, Southern states seceded, resulting in the American Civil War. Pierce was born in New Hampshire, the son of state governor Benjamin Pierce (governor), Benjamin Pierce. He served in the United States House of Representatives, House of Representatives from 1833 until his election to the United States Senate, Senate, where he served from 1837 until his resignation in 1842. Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franklin D
Franklin may refer to: People and characters * Franklin (given name), including list of people and characters with the name * Franklin (surname), including list of people and characters with the name * Franklin (class), a member of a historical English social class Places * Franklin (crater), a lunar impact crater * Franklin County (other), in a number of countries * Mount Franklin (other), including Franklin Mountain Australia * Franklin, Tasmania, a township * Division of Franklin, federal electoral division in Tasmania * Division of Franklin (state), state electoral division in Tasmania * Franklin, Australian Capital Territory, a suburb in the Canberra district of Gungahlin * Franklin River, river of Tasmania * Franklin Sound, waterway of Tasmania Canada * District of Franklin, a former district of the Northwest Territories * Franklin, Quebec, a municipality in the Montérégie region * Rural Municipality of Franklin, Manitoba * Franklin, Manitoba, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Hoover
Herbert Clark Hoover (August 10, 1874 – October 20, 1964) was the 31st president of the United States, serving from 1929 to 1933. A wealthy mining engineer before his presidency, Hoover led the wartime Commission for Relief in Belgium and was the director of the U.S. Food Administration, followed by post-war relief of Europe. As a member of the Republican Party (United States), Republican Party, he served as the third United States secretary of commerce from 1921 to 1928 before being 1928 United States presidential election, elected president in 1928. His presidency was dominated by the Great Depression, and his policies and methods to combat it were seen as lackluster. Amid his unpopularity, he decisively lost reelection to Franklin D. Roosevelt in 1932 United States presidential election, 1932. Born to a Quaker family in West Branch, Iowa, Hoover grew up in Oregon. He was one of the first graduates of the new Stanford University in 1895. Hoover took a position with a Lond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superset
In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique can be seen as a consequence of univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse Relation
In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and L \subseteq X \times Y is a relation from X to Y, then L^ is the relation defined so that yL^x if and only if xLy. In set-builder notation, :L^ = \. Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, the inverse of the original relation,Gerard O'Regan (2016): ''Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications'' or the reciprocal L^ of the relation L. Other notations for the converse relation include L^, L^, \breve, L^, or L^. The notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unordered pair'', denoted , always equals the unordered pair . Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuous Truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" will be true when no cell phones are present in the room. In this case, the statement "all cell phones in the room are turned ''on''" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on ''and'' turned off", which would otherwise be incoherent and false. More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous truths becau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Relation
In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation over ''X''". An example of a homogeneous relation is the relation of kinship, where the relation is between people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric relation, and a general endorelation corresponding to a directed graph. An endorelation ''R'' corresponds to a logical matrix of 0s and 1s, where the expression ''xRy'' (''x'' is ''R''-related to ''y'') corresponds to an edge between ''x'' and ''y'' in the graph, and to a 1 in the square matrix of ''R''. It is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
