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Necessary And Sufficient Conditions
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of (equivalently, it is impossible to have without ). Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" indicate relations betw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourth Of July
Independence Day (colloquially the Fourth of July) is a federal holiday in the United States commemorating the Declaration of Independence, which was ratified by the Second Continental Congress on July 4, 1776, establishing the United States of America. The Founding Father delegates of the Second Continental Congress declared that the Thirteen Colonies were no longer subject (and subordinate) to the monarch of Britain, King George III, and were now united, free, and independent states. The Congress voted to approve independence by passing the Lee Resolution on July 2 and adopted the Declaration of Independence two days later, on July 4. Independence Day is commonly associated with fireworks, parades, barbecues, carnivals, fairs, picnics, concerts, baseball games, family reunions, political speeches, and ceremonies, in addition to various other public and private events celebrating the history, government, and traditions of the United States. Independence Day is the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mammal
Mammals () are a group of vertebrate animals constituting the class Mammalia (), characterized by the presence of mammary glands which in females produce milk for feeding (nursing) their young, a neocortex (a region of the brain), fur or hair, and three middle ear bones. These characteristics distinguish them from reptiles (including birds) from which they diverged in the Carboniferous, over 300 million years ago. Around 6,400 extant species of mammals have been described divided into 29 orders. The largest orders, in terms of number of species, are the rodents, bats, and Eulipotyphla (hedgehogs, moles, shrews, and others). The next three are the Primates (including humans, apes, monkeys, and others), the Artiodactyla ( cetaceans and even-toed ungulates), and the Carnivora (cats, dogs, seals, and others). In terms of cladistics, which reflects evolutionary history, mammals are the only living members of the Synapsida (synapsids); this clade, together with Saur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Intersection
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The number 9 is in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S , assuming that R abbreviates "it is raining" and S abbreviates "it is snowing". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Playing Card
A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a finish to make handling easier. They are most commonly used for playing card games, and are also used in magic tricks, cardistry, card throwing, and card houses; cards may also be collected. Some patterns of Tarot playing card are also used for divination, although bespoke cards for this use are more common. Playing cards are typically palm-sized for convenient handling, and usually are sold together in a set as a deck of cards or pack of cards. The most common type of playing card in the West is the French-suited, standard 52-card pack, of which the most widespread design is the English pattern, followed by the Belgian-Genoese pattern. However, many countries use other, traditional types of playing card, including those that are German ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Veto Override
A veto is a legal power to unilaterally stop an official action. In the most typical case, a president or monarch vetoes a bill to stop it from becoming law. In many countries, veto powers are established in the country's constitution. Veto powers are also found at other levels of government, such as in state, provincial or local government, and in international bodies. Some vetoes can be overcome, often by a supermajority vote: in the United States, a two-thirds vote of the House and Senate can override a presidential veto. Article I, Section 7, Clause 2 of the United States Constitution Some vetoes, however, are absolute and cannot be overridden. For example, in the United Nations Security Council, the permanent members (China, France, Russia, the United Kingdom, and the United States) have an absolute veto over any Security Council resolution. In many cases, the veto power can only be used to prevent changes to the status quo. But some veto powers also include the ability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Veto
A veto is a legal power to unilaterally stop an official action. In the most typical case, a president or monarch vetoes a bill to stop it from becoming law. In many countries, veto powers are established in the country's constitution. Veto powers are also found at other levels of government, such as in state, provincial or local government, and in international bodies. Some vetoes can be overcome, often by a supermajority vote: in the United States, a two-thirds vote of the House and Senate can override a presidential veto. Article I, Section 7, Clause 2 of the United States Constitution Some vetoes, however, are absolute and cannot be overridden. For example, in the United Nations Security Council, the permanent members ( China, France, Russia, the United Kingdom, and the United States) have an absolute veto over any Security Council resolution. In many cases, the veto power can only be used to prevent changes to the status quo. But some veto powers also include the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ICE 3 Fahlenbach
Ice is water frozen into a solid state, typically forming at or below temperatures of 0 degrees Celsius or Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaque bluish-white color. In the Solar System, ice is abundant and occurs naturally from as close to the Sun as Mercury to as far away as the Oort cloud objects. Beyond the Solar System, it occurs as interstellar ice. It is abundant on Earth's surfaceparticularly in the polar regions and above the snow lineand, as a common form of precipitation and deposition, plays a key role in Earth's water cycle and climate. It falls as snowflakes and hail or occurs as frost, icicles or ice spikes and aggregates from snow as glaciers and ice sheets. Ice exhibits at least eighteen phases ( packing geometries), depending on temperature and pressure. When water is cooled rapidly (quenching), up to three types of amorphous ice can form depending on its hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunction (logic)
In logic, mathematics and linguistics, And (\wedge) is the truth function, truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this operator is typically written as \wedge or . A \land B is true if and only if A is true and B is true, otherwise it is false. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English language, English "and". * In programming languages, the short-circuit evaluation, short-circuit and control structure. * In set theory, intersection (set theory), intersection. * In Lattice (order), lattice theory, logical conjunction (greatest lower bound). * In predicate logic, universal quantification. Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
