In logic , extensionality , or extensional equality , refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality , which is concerned with whether the internal definitions of objects are the same.
41-413: The extensional definition of function equality, discussed above, is commonly used in mathematics. A similar extensional definition is usually employed for relations : two relations are said to be equal if they have the same extensions . In set theory , the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it
82-416: A heterogeneous relation R over X and Y is a subset of { ( x , y ) | x ∈ X , y ∈ Y } . When X = Y , the relation concept described above is obtained; it is often called homogeneous relation (or endorelation ) to distinguish it from its generalization. The above properties and operations that are marked " " and " ", respectively, generalize to heterogeneous relations. An example of
123-676: A beautiful thing, a thing I dare not hope if we could spend our life near each other, hypnotized by our dreams: your patriotic dream, our humanitarian dream, and our scientific dream. [Pierre Curie to Maria Skłodowska] The Curies had a happy, affectionate marriage, and they were known for their devotion to each other. Before his famous doctoral studies on magnetism, he designed and perfected an extremely sensitive torsion balance for measuring magnetic coefficients. Variations on this equipment were commonly used by future workers in that area. Pierre Curie studied ferromagnetism , paramagnetism , and diamagnetism for his doctoral thesis, and discovered
164-471: A certain degree" – either they are in relation or they are not. Formally, a relation R over a set X can be seen as a set of ordered pairs ( x , y ) of members of X . The relation R holds between x and y if ( x , y ) is a member of R . For example, the relation " is less than " on the natural numbers is an infinite set R less of pairs of natural numbers that contains both (1,3) and (3,4) , but neither (3,1) nor (4,4) . The relation "
205-463: A heterogeneous relation is "ocean x borders continent y ". The best-known examples are functions with distinct domains and ranges, such as sqrt : N → R + . Pierre Curie Pierre Curie ( / ˈ k jʊər i / KURE -ee ; French: [pjɛʁ kyʁi] ; 15 May 1859 – 19 April 1906) was a French physicist and a pioneer in crystallography , magnetism , piezoelectricity and radioactivity . In 1903, he received
246-507: A mere spectator, and his goal certainly was not to communicate with spirits. He saw the séances as scientific experiments, tried to monitor different parameters, and took detailed notes of every observation. Curie considered himself as atheist . Pierre Curie's grandfather, Paul Curie (1799–1853), a doctor of medicine, was a committed Malthusian humanist and married Augustine Hofer, daughter of Jean Hofer and great-granddaughter of Jean-Henri Dollfus, great industrialists from Mulhouse in
287-464: A nontrivial divisor of" , and, most popular " = " for "is equal to" . For example, " 1 < 3 ", " 1 is less than 3 ", and " (1,3) ∈ R less " mean all the same; some authors also write " (1,3) ∈ (<) ". Various properties of relations are investigated. A relation R is reflexive if xRx holds for all x , and irreflexive if xRx holds for no x . It is symmetric if xRy always implies yRx , and asymmetric if xRy implies that yRx
328-408: A random mixture of sand in zero gravity has no dissymmetry (it is isotropic ). Introduce a gravitational field , and there is a dissymmetry because of the direction of the field. Then the sand grains can 'self-sort' with the density increasing with depth. But this new arrangement, with the directional arrangement of sand grains, actually reflects the dissymmetry of the gravitational field that causes
369-467: A set which are related by an equivalence relation belong to the same equivalence class . Type-theoretical foundations of mathematics are generally not extensional in this sense, and setoids are commonly used to maintain a difference between intensional equality and a more general equivalence relation (which generally has poor constructibility or decidability properties). There are various extensionality principles in mathematics. Depending on
410-446: A sister of herself?), " is ancestor of " is transitive, while " is parent of " is not. Mathematical theorems are known about combinations of relation properties, such as "a transitive relation is irreflexive if, and only if, it is asymmetric". Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is reflexive, antisymmetric, and transitive, an equivalence relation
451-439: A town has one person named Joe, who is also the oldest person in the town. Then, the two predicates "being called Joe", and "being the oldest person in this town" are intensionally distinct, but extensionally equal for the (current) population of this town. Relation (mathematics) In mathematics , a relation denotes some kind of relationship between two objects in a set , which may or may not hold. As an example, "
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#1732884359893492-412: Is R -related to y " and is written in infix notation as xRy . The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy . For example, 3 divides 9 , but 9 does not divide 3 . A relation R on a finite set X may be represented as: A transitive relation R on a finite set X may be also represented as For example, on
533-497: Is a nontrivial divisor of " on the set of one-digit natural numbers is sufficiently small to be shown here: R dv = { (2,4), (2,6), (2,8), (3,6), (3,9), (4,8) } ; for example 2 is a nontrivial divisor of 8 , but not vice versa, hence (2,8) ∈ R dv , but (8,2) ∉ R dv . If R is a relation that holds for x and y one often writes xRy . For most common relations in mathematics, special symbols are introduced, like " < " for "is less than" , and " | " for "is
574-586: Is a professor of nuclear physics at the University of Paris , and their grandson, Pierre Joliot , who was named after Pierre Curie, is a noted biochemist. Pierre Curie died in a street collision in Paris on 19 April 1906. Crossing the busy Rue Dauphine in the rain at the Quai de Conti, he slipped and fell under a heavy horse-drawn cart. One of the wheels ran over his head, fracturing his skull and killing him instantly. Both
615-591: Is a relation that is reflexive, symmetric, and transitive, a function is a relation that is right-unique and left-total (see below ). Since relations are sets, they can be manipulated using set operations, including union , intersection , and complementation , leading to the algebra of sets . Furthermore, the calculus of relations includes the operations of taking the converse and composing relations . The above concept of relation has been generalized to admit relations between members of two different sets ( heterogeneous relation , like " lies on " between
656-541: Is a unit of measurement (3.7 × 10 decays per second or 37 gigabecquerels ) used to describe the intensity of a sample of radioactive material and was named after Marie and Pierre Curie by the Radiology Congress in 1910. Pierre Curie formulated what is now known as the Curie Dissymmetry Principle : a physical effect cannot have a dissymmetry absent from its efficient cause . For example,
697-430: Is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same extension to be distinguished. Other mathematical objects are also constructed in such a way that the intuitive notion of "equality" agrees with set-level extensional equality; thus, equal ordered pairs have equal elements, and elements of
738-521: Is contained in S and S is contained in R , then R and S are called equal written R = S . If R is contained in S but S is not contained in R , then R is said to be smaller than S , written R ⊊ S . For example, on the rational numbers , the relation > is smaller than ≥ , and equal to the composition > ∘ > . The above concept of relation has been generalized to admit relations between members of two different sets. Given sets X and Y ,
779-403: Is impossible. It is transitive if xRy and yRz always implies xRz . For example, " is less than " is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric. " is sister of " is transitive, but neither reflexive (e.g. Pierre Curie is not a sister of himself), nor symmetric, nor asymmetric; while being irreflexive or not may be a matter of definition (is every woman
820-526: Is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the values 3 and 1 nor between 4 and 4 , that is, 3 < 1 and 4 < 4 both evaluate to false. As another example, " is sister of " is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska , and likewise vice versa. Set members may not be in relation "to
861-683: The Nobel Prize in Physics with his wife, Marie Curie , and Henri Becquerel , "in recognition of the extraordinary services they have rendered by their joint researches on the radiation phenomena discovered by Professor Henri Becquerel". With their win, the Curies became the first married couple to win the Nobel Prize, launching the Curie family legacy of five Nobel Prizes. Born in Paris on 15 May 1859, Pierre Curie
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#1732884359893902-741: The Sorbonne , also known as the University of Paris. He did not proceed immediately to a doctorate due to lack of money. Instead, he worked as a laboratory instructor. When Pierre Curie was preparing for his Bachelor of Science degree, he worked in the laboratory of Jean-Gustave Bourbouze in the Faculty of Science. In 1895, he went on to receive his doctorate at the University of Paris . The submission material for his doctorate consisted of his research over magnetism . After obtaining his doctorate, he became professor of physics and in 1900, he became professor in
943-655: The Curies experienced radium burns, both accidentally and voluntarily, and were exposed to extensive doses of radiation while conducting their research. They experienced radiation sickness and Marie Curie died from radiation-induced aplastic anemia in 1934. Even now, all their papers from the 1890s, even her cookbooks, are too dangerous to touch. Their laboratory books are kept in special lead boxes and people who want to see them have to wear protective clothing . Most of these items can be found at Bibliothèque nationale de France . Had Pierre Curie not been killed in an accident as he was, he would most likely have eventually died of
984-414: The above properties are particularly useful, and thus have received names by their own. Orderings: Uniqueness properties: Uniqueness and totality properties: A relation R over sets X and Y is said to be contained in a relation S over X and Y , written R ⊆ S , if R is a subset of S , that is, for all x ∈ X and y ∈ Y , if xRy , then xSy . If R
1025-474: The chosen foundation, some extensionality principles may imply another. For example it is well known that in univalent foundations , the univalence axiom implies both propositional and functional extensionality. Extensionality principles are usually assumed as axioms, especially in type theories where computational content must be preserved. However, in set theory and other extensional foundations, functional extensionality can be proven to hold by default. Consider
1066-500: The diagram below is neither irreflexive, nor reflexive, since it contains the pair (0,0) , but not (2,2) , respectively. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by x > 2 is neither symmetric (e.g. 5 R 1 , but not 1 R 5 ) nor antisymmetric (e.g. 6 R 4 , but also 4 R 6 ), let alone asymmetric. Uniqueness properties: Totality properties: Relations that satisfy certain combinations of
1107-559: The effect of temperature on paramagnetism which is now known as Curie's law . The material constant in Curie's law is known as the Curie constant . He also discovered that ferromagnetic substances exhibited a critical temperature transition, above which the substances lost their ferromagnetic behavior. This is now known as the Curie temperature . The Curie temperature is used to study plate tectonics, treat hypothermia, measure caffeine, and to understand extraterrestrial magnetic fields. The Curie
1148-465: The faculty of sciences. In 1880, Pierre and his older brother Paul-Jacques (1856–1941) demonstrated that an electric potential was generated when crystals were compressed, i.e., piezoelectricity . To aid this work they invented the piezoelectric quartz electrometer. The following year they demonstrated the reverse effect: that crystals could be made to deform when subject to an electric field. Almost all digital electronic circuits now rely on this in
1189-401: The first discovery of nuclear energy , by identifying the continuous emission of heat from radium particles. Curie also investigated the radiation emissions of radioactive substances, and through the use of magnetic fields was able to show that some of the emissions were positively charged, some were negative and some were neutral. These correspond to alpha , beta and gamma radiation . In
1230-569: The form of crystal oscillators . In subsequent work on magnetism Pierre Curie defined the Curie scale. This work also involved delicate equipment – balances, electrometers, etc. Pierre Curie was introduced to Maria Skłodowska by their friend, physicist Józef Wierusz-Kowalski . Curie took her into his laboratory as his student. His admiration for her grew when he realized that she would not inhibit his research. He began to regard Skłodowska as his muse. She refused his initial proposal, but finally agreed to marry him on 26 July 1895. It would be
1271-756: The late nineteenth century, Pierre Curie was investigating the mysteries of ordinary magnetism when he became aware of the spiritualist experiments of other European scientists, such as Charles Richet and Camille Flammarion . Pierre Curie initially thought the systematic investigation into the paranormal could help with some unanswered questions about magnetism. He wrote to Marie, then his fiancée: "I must admit that those spiritual phenomena intensely interest me. I think they are questions that deal with physics." Pierre Curie's notebooks from this period show he read many books on spiritualism. He did not attend séances such as those of Eusapia Palladino in Paris in June 1905 as
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1312-587: The lecture so Lord Kelvin sat beside her while Pierre spoke on their research. After this, Lord Kelvin held a luncheon for Pierre. While in London, Pierre and Marie were awarded the Davy Medal of the Royal Society of London. In the same year, Pierre and Marie Curie, as well as Henri Becquerel, were awarded a Nobel Prize in physics for their research of radioactivity. Curie and one of his students, Albert Laborde, made
1353-433: The relation R el on R by The representation of R el as a 2D-plot obtains an ellipse, see right picture. Since R is not finite, neither a directed graph, nor a finite Boolean matrix, nor a Hasse diagram can be used to depict R el . Some important properties that a relation R over a set X may have are: The previous 2 alternatives are not exhaustive; e.g., the red relation y = x given in
1394-478: The second half of the 18th century and the first part of the 19th century. Through this paternal grandmother, Pierre Curie is also a direct descendant of the Basel scientist and mathematician Jean Bernoulli (1667–1748), as is Pierre-Gilles de Gennes , winner of the 1991 Nobel Prize in Physics. Pierre and Marie Curie's daughter, Irène , and their son-in-law, Frédéric Joliot-Curie , were also physicists involved in
1435-459: The separation. Curie worked with his wife in isolating polonium and radium . They were the first to use the term " radioactivity ", and were pioneers in its study. Their work, including Marie Curie's celebrated doctoral work, made use of a sensitive piezoelectric electrometer constructed by Pierre and his brother Jacques Curie. Pierre Curie's 26 December 1898 publication with his wife and M. G. Bémont for their discovery of radium and polonium
1476-513: The set of all points and that of all lines in geometry), relations between three or more sets ( finitary relation , like "person x lives in town y at time z " ), and relations between classes (like " is an element of " on the class of all sets, see Binary relation § Sets versus classes ). Given a set X , a relation R over X is a set of ordered pairs of elements from X , formally: R ⊆ { ( x , y ) | x , y ∈ X } . The statement ( x , y ) ∈ R reads " x
1517-466: The set of all divisors of 12 , define the relation R div by Formally, X = { 1, 2, 3, 4, 6, 12 } and R div = { (1,2), (1,3), (1,4), (1,6), (1,12), (2,4), (2,6), (2,12), (3,6), (3,12), (4,12), (6,12) } . The representation of R div as a Boolean matrix is shown in the middle table; the representation both as a Hasse diagram and as a directed graph is shown in the left picture. The following are equivalent: As another example, define
1558-407: The study of radioactivity , and each also received Nobel prizes for their work. The Curies' other daughter, Ève , wrote a noted biography of her mother. She was the only member of the Curie family to not become a physicist. Ève married Henry Richardson Labouisse Jr. , who received a Nobel Peace Prize on behalf of UNICEF in 1965. Pierre and Marie Curie's granddaughter, Hélène Langevin-Joliot ,
1599-479: The two functions f and g mapping from and to natural numbers , defined as follows: These functions are extensionally equal; given the same input, both functions always produce the same value. But the definitions of the functions are not equal, and in that intensional sense the functions are not the same. Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionally identical. For example, suppose that
1640-650: Was honored by a Citation for Chemical Breakthrough Award from the Division of History of Chemistry of the American Chemical Society presented to the ESPCI ParisTech (officially the École supérieure de physique et de Chimie industrielles de la Ville de Paris) in 2015. In 1903, to honor the Curies' work, the Royal Society of London invited Pierre to present their research. Marie Curie was not permitted to give
1681-424: Was the son of Eugène Curie (1827–1910), a doctor of French Huguenot Protestant origin from Alsace , and Sophie-Claire Curie (née Depouilly; 1832–1897). He was educated by his father and in his early teens showed a strong aptitude for mathematics and geometry. When he was 16, he earned his Bachelor of Science in mathematics. By the age of 18, he earned his license in physical sciences from the Faculty of Sciences at