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Negated AND Gate
In digital electronics , a NAND GATE (NEGATIVE-AND) is a logic gate which produces an output which is false only if all its inputs are true; thus its output is complement to that of the AND gate . A LOW (0) output results only if both the inputs to the gate are HIGH (1); if one or both inputs are LOW (0), a HIGH (1) output results. It is made using transistors and junction diodes. By De Morgan\'s theorem , AB=A+B, and thus a N AND gate is equivalent to inverters followed by an OR gate . The N AND 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(a1, a2, ..., an) is logically equivalent to NOT(a1 AND a2 AND ... AND an). CONTENTS * 1 Symbols * 2 Hardware description and pinout * 3 Implementations * 3.1 Alternatives * 4 See also * 5 References * 6 External links SYMBOLSThere 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 N AND gate is a standard AND gate with an inversion bubble connected
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Digital Electronics
DIGITAL ELECTRONICS or DIGITAL (ELECTRONIC) CIRCUITS are electronics that handle digital signals (discrete bands of analog levels) rather than by continuous ranges as used in analog electronics . All levels within a band of values represent the same information state . Because of this discretization , relatively small changes to the analog signal levels due to manufacturing tolerance , signal attenuation or noise do not leave the discrete envelope, and as a result are ignored by signal state sensing circuitry. In most cases, the number of these states is two, and they are represented by two voltage bands: one near a reference value (typically termed as "ground" or zero volts), and the other a value near the supply voltage. These correspond to the _false_ and _true_ values of the Boolean domain respectively. Digital techniques are useful because it is easier to get an electronic device to switch into one of a number of known states than to accurately reproduce a continuous range of values. Digital electronic circuits are usually made from large assemblies of logic gates , simple electronic representations of Boolean logic functions . CONTENTS * 1 History * 2 Properties * 3 Construction * 4 Design * 4.1 Structure of digital systems * 4.1.1 Representation * 4.1.2 Combinational vs
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Logic Gate
In electronics , a LOGIC GATE is an idealized or physical device implementing a Boolean function ; that is, it performs a logical operation on one or more binary inputs and produces a single binary output. Depending on the context, the term may refer to an IDEAL LOGIC GATE, one that has for instance zero rise time and unlimited fan-out , or it may refer to a non-ideal physical device (see Ideal and real op-amps for comparison). Logic gates are primarily implemented using diodes or transistors acting as electronic switches , but can also be constructed using vacuum tubes , electromagnetic relays (relay logic ), fluidic logic , pneumatic logic , optics , molecules , or even mechanical elements. With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing the construction of a physical model of all of Boolean logic , and therefore, all of the algorithms and mathematics that can be described with Boolean logic. Logic circuits include such devices as multiplexers , registers , arithmetic logic units (ALUs), and computer memory , all the way up through complete microprocessors , which may contain more than 100 million gates. In modern practice, most gates are made from field-effect transistors (FETs), particularly metal–oxide–semiconductor field-effect transistors (MOSFETs)
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Complement (set Theory)
In set theory , the COMPLEMENT of a set A refers to elements not in A. The RELATIVE COMPLEMENT of A with respect to a set B, also termed the DIFFERENCE of sets A and B, written B ∖ A, is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the ABSOLUTE COMPLEMENT of A is the set of elements in U but not in A. CONTENTS* 1 Relative complement * 1.1 Definition * 1.2 Examples * 1.3 Properties * 2 Absolute complement * 2.1 Definition * 2.2 Examples * 2.3 Properties * 3 LaTeX notation * 4 Complements in various programming languages * 5 See also * 6 Notes * 7 References * 8 External links RELATIVE COMPLEMENTDEFINITIONIf A and B are sets, then the RELATIVE COMPLEMENT of A in B, also termed the SET-THEORETIC DIFFERENCE of B and A, is the set of elements in B but not in A. The RELATIVE COMPLEMENT of A (left circle) in B (right circle): B A = B A {displaystyle Bcap A^{complement }=Bsetminus A} The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard . It is sometimes written B − A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements b − a, where b is taken from B and a from A. Formally: B A = { x B x A }
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AND Gate
The AND GATE is a basic digital logic gate that implements logical conjunction - it behaves according to the truth table to the right. A HIGH output (1) results only if both the inputs to the AND gate are HIGH (1). If neither or only one input to the AND gate is HIGH, a LOW output results. In another sense, the function of AND effectively finds the _minimum_ between two binary digits, just as the OR function finds the _maximum_ between two binary digits. Therefore, the output is always 0, except when all the inputs are 1. CONTENTS * 1 Symbols * 2 Implementations * 2.1 Alternatives * 3 IC package * 4 See also * 5 References SYMBOLSThere are three symbols for AND gates: the American (ANSI or 'military') symbol and the IEC ('European' or 'rectangular') symbol, as well as the deprecated DIN symbol. For more information see Logic Gate Symbols . _ MIL/ANSI Symbol_ _IEC Symbol_ _DIN Symbol_The AND gate with inputs _A_ and _B_ and output _C_ implements the logical expression C = A B {displaystyle C=Acdot B} . IMPLEMENTATIONS AND gate using diodes AND gate using transistors NMOS AND gate An AND gate is usually designed using N-channel (pictured) or P-channel MOSFETs . The digital inputs A and B cause the output F to have the same result as the AND function
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De Morgan's Theorem
In propositional logic and boolean algebra , DE MORGAN\'S LAWS are a pair of transformation rules that are both valid rules of inference . They are named after Augustus De Morgan
Augustus De Morgan
, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation . The rules can be expressed in English as: the negation of a disjunction is the conjunction of the negations; and the negation of a conjunction is the disjunction of the negations; or the complement of the union of two sets is the same as the intersection of their complements; and the complement of the intersection of two sets is the same as the union of their complements
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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. CONTENTS* 1 Electronic implementation * 1.1 Digital building block * 1.2 Alternatives * 1.3 Performance measurement * 2 See also * 3 External links * 4 References ELECTRONIC IMPLEMENTATION* NMOS inverter * PMOS inverter * Static CMOS
CMOS
inverter * NPN transistor–transistor logic inverter * Depletion-load NMOS logic
Depletion-load NMOS logic
NAND * Saturated-load NMOS inverter * NPN resistor–transistor logic inverter 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 then 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
CMOS
configuration
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OR Gate
The OR GATE is a digital logic gate that implements logical disjunction – it behaves according to the truth table to the right. A HIGH output (1) results if one or both the inputs to the gate are HIGH (1). If neither input is high, a LOW output (0) results. In another sense, the function of OR effectively finds the maximum between two binary digits, just as the complementary AND function finds the minimum. CONTENTS * 1 Symbols * 2 Hardware description and pinout * 3 Implementations * 3.1 Alternatives * 4 Wired-OR * 5 See also * 6 References SYMBOLSThere are two symbols of OR gates: the American (ANSI or 'military') symbol and the IEC ('European' or 'rectangular') symbol, as well as the deprecated DIN
DIN
symbol. For more information see Logic Gate Symbols . MIL/ANSI Symbol IEC Symbol DIN
DIN
Symbol This schematic diagram shows the arrangement of four OR gates within a standard 4071 CMOS
CMOS
integrated circuit. HARDWARE DESCRIPTION AND PINOUTOR Gates are basic logic gates, and as such they are available in TTL and CMOS
CMOS
ICs logic families . The standard 4000 series CMOS
CMOS
IC is the 4071, which includes four independent two-input OR gates. The ancestral TTL device is the 7432
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Boolean Function
In mathematics and logic , a (FINITARY ) BOOLEAN FUNCTION (or switching function) is a function of the form _ƒ_ : B_k_ → B, where B = {0, 1} is a _ Boolean domain _ and _k_ is a non-negative integer called the arity of the function. In the case where _k_ = 0, the "function" is essentially a constant element of B. Every _k_-ary Boolean function can be expressed as a propositional formula in _k_ variables _x_1, …, _x__k_, and two propositional formulas are logically equivalent if and only if they express the same Boolean function. There are 22_k_ _k_-ary functions for every _k_. BOOLEAN FUNCTIONS IN APPLICATIONSA Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. Such functions play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers . The properties of Boolean functions play a critical role in cryptography , particularly in the design of symmetric key algorithms (see substitution box ). Boolean functions are often represented by sentences in propositional logic , and sometimes as multivariate polynomials over GF (2), but more efficient representations are binary decision diagrams (BDD), negation normal forms , and propositional directed acyclic graphs (PDAG)
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Functional Completeness
In logic , a FUNCTIONALLY COMPLETE set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression . A well-known complete set of connectives is { AND, NOT }, consisting of binary conjunction and negation . Each of the singleton sets { NAND } and { NOR } is functionally complete. In a context of propositional logic , functionally complete sets of connectives are also called (EXPRESSIVELY) ADEQUATE. From the point of view of digital electronics , functional completeness means that every possible logic gate can be realized as a network of gates of the types prescribed by the set. In particular, all logic gates can be assembled from either only binary NAND gates , or only binary NOR gates . CONTENTS * 1 Introduction * 2 Formal definition * 3 Characterization of functional completeness * 4 Minimal functionally complete operator sets * 5 Examples * 6 In other domains * 7 Set theory * 8 See also * 9 References INTRODUCTIONModern texts on logic typically take as primitive some subset of the connectives: conjunction ( {displaystyle land } ); disjunction ( {displaystyle lor } ); negation ( {displaystyle neg } ); material conditional ( {displaystyle to } ); and possibly the biconditional ( {displaystyle leftrightarrow } )
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NOR Gate
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 (1) results if both the inputs to the gate are LOW (0); if one or both input is HIGH (1), a LOW output (0) results. NOR is the result of the negation of the OR operator. It can also 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_ as it can only change LOW to HIGH but not vice versa. 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; it may use a NOR followed by a NOT. A significant exception is some forms of the domino logic family. The original Apollo Guidance Computer used 4,100 ICs, each one containing only a single 3-input NOR gate. CONTENTS * 1 Symbols * 2 Hardware description and pinout * 2.1 Availability * 3 Implementations * 3.1 Alternatives * 4 See also * 5 References * 6 External links SYMBOLSThere are three symbols for NOR gates: the American (ANSI or 'military') symbol and the IEC ('European' or 'rectangular') symbol, as well as the deprecated DIN symbol. For more information see Logic Gate Symbols
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Logical Equivalence
In logic , statements p {displaystyle p} and q {displaystyle q} are LOGICALLY EQUIVALENT if they have the same logical content. This is a semantic concept; two statements are equivalent if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of p {displaystyle p} and q {displaystyle q} is sometimes expressed as p q {displaystyle pequiv q} , E p q {displaystyle {textsf {E}}pq} , or p q {displaystyle piff q} . However, these symbols are also used for material equivalence; the proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related
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ANSI
The AMERICAN NATIONAL STANDARDS INSTITUTE (ANSI, /ˈænsi/ AN-see ) is a private non-profit organization that oversees the development of voluntary consensus standards for products, services, processes, systems, and personnel in the United States. The organization also coordinates U.S. standards with international standards so that American products can be used worldwide. ANSI accredits standards that are developed by representatives of other standards organizations , government agencies , consumer groups , companies, and others. These standards ensure that the characteristics and performance of products are consistent, that people use the same definitions and terms, and that products are tested the same way. ANSI also accredits organizations that carry out product or personnel certification in accordance with requirements defined in international standards. The organization's headquarters are in Washington, D.C. ANSI's operations office is located in New York City
New York City
. The ANSI annual operating budget is funded by the sale of publications, membership dues and fees, accreditation services, fee-based programs, and international standards programs
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International Electrotechnical Commission
The INTERNATIONAL ELECTROTECHNICAL COMMISSION (IEC; in French : _Commission électrotechnique internationale_) is a non-profit , non-governmental international standards organization that prepares and publishes International Standards for all electrical , electronic and related technologies – collectively known as "electrotechnology ". IEC standards cover a vast range of technologies from power generation, transmission and distribution to home appliances and office equipment, semiconductors, fibre optics, batteries, solar energy , nanotechnology and marine energy as well as many others. The IEC also manages three global conformity assessment systems that certify whether equipment, system or components conform to its International Standards. The IEC charter embraces all electrotechnologies including energy production and distribution, electronics, magnetics and electromagnetics , electroacoustics , multimedia , telecommunication and medical technology , as well as associated general disciplines such as terminology and symbols, electromagnetic compatibility (by its Advisory Committee on Electromagnetic Compatibility , ACEC), measurement and performance, dependability, design and development, safety and the environment
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DIN
_DEUTSCHES INSTITUT FüR NORMUNG E.V._ (DIN; in English , the GERMAN INSTITUTE FOR STANDARDIZATION) is the German national organization for standardization and is the German ISO member body. DIN is a German Registered Association (_e.V. _) headquartered in Berlin . There are currently around thirty thousand DIN Standards , covering nearly every field of technology. CONTENTS * 1 History * 2 DIN standard designation * 3 Examples of DIN standards * 4 See also * 5 External links HISTORYFounded in 1917 as the _Normenausschuß der deutschen Industrie_ (NADI, "Standardisation Committee of German Industry"), the NADI was renamed _Deutscher Normenausschuß_ (DNA, "German Standardisation Committee") in 1926 to reflect that the organization now dealt with standardization issues in many fields; viz., not just for industrial products. In 1975 it was renamed again to _Deutsches Institut für Normung_, or 'DIN' and is recognized by the German government as the official national-standards body, representing German interests at the international and European levels. The acronym , 'DIN' is often incorrectly expanded as _Deutsche Industrienorm_ ("German Industry Standard"). This is largely due to the historic origin of the DIN as "NADI". The NADI indeed published their standards as _DI-Norm_ (_Deutsche Industrienorm_). For example, the first published standard was 'DI-Norm 1' (about tapered pins ) in 1918
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Transistor–transistor Logic
TRANSISTOR–TRANSISTOR LOGIC (TTL) is a class of digital circuits built from bipolar junction transistors (BJTs) and resistors . It is called _transistor–transistor logic_ because transistors perform both the logic function (e.g., AND ) and the amplifying function (compare with resistor–transistor logic (RTL) and diode–transistor logic (DTL)). TTL integrated circuits (ICs) were widely used in applications such as computers , industrial controls, test equipment and instrumentation, consumer electronics, and synthesizers . The designation _TTL_ is sometimes used to mean TTL-compatible logic levels , even when not associated directly with TTL integrated circuits, for example as a label on the inputs and outputs of electronic instruments. After their introduction in integrated circuit form in 1963 by Sylvania , TTL integrated circuits were manufactured by several semiconductor companies. The 7400 series (also called 74xx) by Texas Instruments became particularly popular. TTL manufacturers offered a wide range of logic gate, flip-flops, counters, and other circuits. Several variations of the original TTL circuit design were developed. The variations offered interchangeable functions that had higher speed or lower power dissipation to allow design optimization. TTL devices were originally made in ceramic and plastic dual-in-line (DIP) packages, and flat-pack form. TTL chips are now also made in surface-mount packages
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