The IMPLY gate is a digital logic gate that implements a logical conditional. There are two symbols for IMPLY gates: the IEEE symbol. For more information see Logic Gate Symbols; the logic symbol → can be used to denote IMPLY in algebraic expressions. IMPLY gate can be implemented by two memristors. AND gate NOT gate NAND gate NOR gate XOR gate XNOR gate Boolean algebra Logic gates
The NOR gate is a digital logic gate that implements logical NOR - it behaves according to the truth table to the right. A HIGH output results if both the inputs to the gate are LOW. NOR is the result of the negation of the OR operator, it can be seen as an AND gate with all the inputs inverted. NOR is a functionally complete operation—NOR gates can be combined to generate any other logical function, it shares this property with the NAND gate. By contrast, the OR operator is monotonic. In most, but not all, circuit implementations, the negation comes for free—including CMOS and TTL. In such logic families, OR is the more complicated operation. A significant exception is some forms of the domino logic family; the original Apollo Guidance Computer used 4,100 integrated circuits, each one containing only a single 3-input NOR gate. There are three symbols for NOR gates: the American symbol and the IEC symbol, as well as the deprecated DIN symbol. For more information see Logic Gate Symbols; the ANSI symbol for the NOR gate is a standard OR gate with an inversion bubble connected.
NOR Gates are basic logic gates, as such they are recognised in TTL and CMOS ICs. The standard, 4000 series, CMOS IC is the 4001, which includes four independent, two-input, NOR gates; the pinout diagram is as follows: These devices are available from most semiconductor manufacturers such as Fairchild Semiconductor, Philips or Texas Instruments. These are available in both through-hole DIP and SOIC format. Datasheets are available in most datasheet databases. In the popular CMOS and TTL logic families, NOR gates with up to 8 inputs are available: CMOS 4001: Quad 2-input NOR gate 4025: Triple 3-input NOR gate 4002: Dual 4-input NOR gate 4078: Single 8-input NOR gate TTL 7402: Quad 2-input NOR gate 7427: Triple 3-input NOR gate 7425: Dual 4-input NOR gate 74260: Dual 5-Input NOR Gate 744078: Single 8-input NOR GateIn the older RTL and ECL families, NOR gates were efficient and most used; the diagrams above show the construction of a 2-input NOR gate using NMOS logic circuitry. If either of the inputs is high, the corresponding N-channel MOSFET is turned on and the output is pulled low.
The diagram below shows a 2-input NOR gate using CMOS technology. The diodes and resistors on the inputs are to protect the CMOS components from damage due to electrostatic discharge and play no part in the logical function of the circuit. If no specific NOR gates are available, one can be made from NAND gates, because NAND and NOR gates are considered the "universal gates", meaning that they can be used to make all the other gates. AND gate OR gate NOT gate NAND gate XOR gate XNOR gate Boolean algebra Logic gates NOR logic Flash memory Interactive NOR gate, Displays the logic simulation of the NOR Gate
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Truth is most used to mean being in accord with fact or reality, or fidelity to an original or standard. Truth is sometimes defined in modern contexts as an idea of "truth to self", or authenticity. Truth is held to be opposite to falsehood, correspondingly, can suggest a logical, factual, or ethical meaning; the concept of truth is discussed and debated in several contexts, including philosophy, art and science. Most human activities depend upon the concept, where its nature as a concept is assumed rather than being a subject of discussion; some philosophers view the concept of truth as basic, unable to be explained in any terms that are more understood than the concept of truth itself. To some, truth is viewed as the correspondence of language or thought to an independent reality, in what is sometimes called the correspondence theory of truth. Various theories and views of truth continue to be debated among scholars and theologians. Language is a means; the method used to determine whether something is a truth is termed a criterion of truth.
There are varying stances on such questions as what constitutes truth: what things are truthbearers capable of being true or false. The English word truth is derived from Old English tríewþ, tréowþ, trýwþ, Middle English trewþe, cognate to Old High German triuwida, Old Norse tryggð. Like troth, it is a -th nominalisation of the adjective true; the English word true is from Old English tríewe, tréowe, cognate to Old Saxon trûui, Old High German triuwu, Old Norse tryggr, Gothic triggws, all from a Proto-Germanic *trewwj- "having good faith" ultimately from PIE *dru- "tree", on the notion of "steadfast as an oak". Old Norse trú, "word of honour. Thus,'truth' involves both the quality of "faithfulness, loyalty, veracity", that of "agreement with fact or reality", in Anglo-Saxon expressed by sōþ. All Germanic languages besides English have introduced a terminological distinction between truth "fidelity" and truth "factuality". To express "factuality", North Germanic opted for nouns derived from sanna "to assert, affirm", while continental West Germanic opted for continuations of wâra "faith, pact".
Romance languages use terms following the Latin veritas, while the Greek aletheia, Russian pravda and South Slavic istina have separate etymological origins. The question of what is a proper basis for deciding how words, symbols and beliefs may properly be considered true, whether by a single person or an entire society, is dealt with by the five most prevalent substantive theories of truth listed below; each presents perspectives that are shared by published scholars. Theories other than the most prevalent substantive theories are discussed. More developed "deflationary" or "minimalist" theories of truth have emerged as possible alternatives to the most prevalent substantive theories. Minimalist reasoning centres around the notion that the application of a term like true to a statement does not assert anything significant about it, for instance, anything about its nature. Minimalist reasoning realises truth as a label utilised in general discourse to express agreement, to stress claims, or to form general assumptions.
Correspondence theories emphasise that true beliefs and true statements correspond to the actual state of affairs. This type of theory stresses a relationship between thoughts or statements on one hand, things or objects on the other, it is a traditional model tracing its origins to ancient Greek philosophers such as Socrates and Aristotle. This class of theories holds that the truth or the falsity of a representation is determined in principle by how it relates to "things", by whether it describes those "things." A classic example of correspondence theory is the statement by the thirteenth century philosopher and theologian Thomas Aquinas: "Veritas est adaequatio rei et intellectus", which Aquinas attributed to the ninth century Neoplatonist Isaac Israeli. Aquinas restated the theory as: "A judgment is said to be true when it conforms to the external reality". Correspondence theory centres around the assumption that truth is a matter of copying what is known as "objective reality" and representing it in thoughts and other symbols.
Many modern theorists have stated that this ideal cannot be achieved without analysing additional factors. For example, language plays a role in that all languages have words to represent concepts that are undefined in other languages; the German word Zeitgeist is one such example: one who speaks or understands the language may "know" what it means, but any translation of the word fails to capture its full meaning. Thus, some words add an additional parameter to the construction of an accurate truth predicate. Among the philosophers who grappled with this problem is Alfred Tarski, whose semantic theory is summarized further below in this article. Proponents of several of the theories below have gone further to a
Inverter (logic gate)
In digital logic, an inverter or NOT gate is a logic gate which implements logical negation. The truth table is shown on the right. An inverter circuit outputs a voltage representing the opposite logic-level to its input, its main function is to invert the input signal applied. If the applied input is low the output becomes high and vice versa. Inverters can be constructed using a single NMOS transistor or a single PMOS transistor coupled with a resistor. Since this'resistive-drain' approach uses only a single type of transistor, it can be fabricated at low cost. However, because current flows through the resistor in one of the two states, the resistive-drain configuration is disadvantaged for power consumption and processing speed. Alternatively, inverters can be constructed using two complementary transistors in a CMOS configuration; this configuration reduces power consumption since one of the transistors is always off in both logic states. Processing speed can be improved due to the low resistance compared to the NMOS-only or PMOS-only type devices.
Inverters can be constructed with bipolar junction transistors in either a resistor–transistor logic or a transistor–transistor logic configuration. Digital electronics circuits operate at fixed voltage levels corresponding to a logical 0 or 1. An inverter circuit serves. Implementation determines the actual voltage; the inverter is a basic building block in digital electronics. Multiplexers, state machines, other sophisticated digital devices may use inverters; the hex inverter is an integrated circuit. For example, the 7404 TTL chip which has 14 pins and the 4049 CMOS chip which has 16 pins, 2 of which are used for power/referencing, 12 of which are used by the inputs and outputs of the six inverters. F = 1 − a is the analytical representation of NOT gate: f = 1 − 0 = 1 f = 1 − 1 = 0 If no specific NOT gates are available, one can be made from the universal NAND or NOR gates. Digital inverter quality is measured using the voltage transfer curve, a plot of output vs. input voltage. From such a graph, device parameters including noise tolerance and operating logic levels can be obtained.
Ideally, the VTC appears as an inverted step function – this would indicate precise switching between on and off – but in real devices, a gradual transition region exists. The VTC indicates; the slope of this transition region is a measure of quality – steep slopes yield precise switching. The tolerance to noise can be measured by comparing the minimum input to the maximum output for each region of operation. Controlled NOT gate AND gate OR gate NAND gate NOR gate XOR gate XNOR gate IMPLY gate Boolean algebra Logic gate The Not Gate on All About Circuits Datasheet: CMOS Hex Buffer/Converter
In digital electronics, a NAND gate is a logic gate which produces an output, false only if all its inputs are true. A LOW output results only if all the inputs to the gate are HIGH. A NAND gate is made using transistors and junction diodes. By De Morgan's theorem, a two-input NAND gate's logic may be expressed as AB=A+B, making a NAND gate equivalent to inverters followed by an OR gate; the NAND gate is significant because any boolean function can be implemented by using a combination of NAND gates. This property is called functional completeness, it shares this property with the NOR gate. Digital systems employing certain logic circuits take advantage of NAND's functional completeness; the function NAND is logically equivalent to NOT. One way of expressing A NAND B is A ∧ B ¯, where the symbol ∧ signifies AND and the bar signifies the negation of the expression under it: in essence ¬. There are three symbols for NAND gates: the MIL/ANSI symbol, the IEC symbol and the deprecated DIN symbol sometimes found on old schematics.
For more information see logic gate symbols. The ANSI symbol for the NAND gate is a standard AND gate with an inversion bubble connected. NAND gates are basic logic gates, as such they are recognised in TTL and CMOS ICs; the NAND gate has the property of functional completeness. That is, any other logic function can be implemented using only NAND gates. An entire processor can be created using NAND gates alone. In TTL ICs using multiple-emitter transistors, it requires fewer transistors than a NOR gate. If no specific NAND gates are available, one can be made from NOR gates, because NAND and NOR gates are considered the "universal gates", meaning that they can be used to make all the other gates. AND gate OR gate NOT gate NOR gate XOR gate XNOR gate Boolean algebra Logic gate NAND logic Digital electronics Flash memory TTL NAND and AND gates – All About Circuits
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves circles, each representing a set; the points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to read visualizations. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets, they are thus a special case of Euler diagrams, which do not show all relations. Venn diagrams were conceived around 1880 by John Venn, they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, statistics and computer science. A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram.
This example involves A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures; the blue circle, set B, represents the living creatures. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that both can fly and have two legs—for example, parrots—are in both sets, so they correspond to points in the region where the blue and orange circles overlap, it is important to note that this overlapping region would only contain those elements that are members of both set A and are members of set B Humans and penguins are bipedal, so are in the orange circle, but since they cannot fly they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles.
The combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that can fly; the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. Venn diagrams were introduced in 1880 by John Venn in a paper entitled On the Diagrammatic and Mechanical Representation of Propositions and Reasonings in the "Philosophical Magazine and Journal of Science", about the different ways to represent propositions by diagrams; the use of these types of diagrams in formal logic, according to Frank Ruskey and Mark Weston, is "not an easy history to trace, but it is certain that the diagrams that are popularly associated with Venn, in fact, originated much earlier. They are rightly associated with Venn, because he comprehensively surveyed and formalized their usage, was the first to generalize them".
Venn himself did not use the term "Venn diagram" and referred to his invention as "Eulerian Circles". For example, in the opening sentence of his 1880 article Venn writes, "Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that called'Eulerian circles,' has met with any general acceptance..." Lewis Carroll includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book "Symbolic Logic". The term "Venn diagram" was used by Clarence Irving Lewis in 1918, in his book "A Survey of Symbolic Logic". Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. M. E. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler, but much of it was unpublished.
She observes earlier Euler-like diagrams by Ramon Llull in the 13th Century. In the 20th century, Venn diagrams were further developed. D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number, he showed that such symmetric Venn diagrams exist when n is five or seven. In 2002 Peter Hamburger found symmetric Venn diagrams for n = 11 and in 2003, Griggs and Savage showed that symmetric Venn diagrams exist for all other primes, thus rotationally symmetric Venn diagrams exist. Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since they have been adopted in the curriculum of other fields such as reading. A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis, the "principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.
That is, the diagram leaves room for any possible relation