NAND and logic gates are the two pillars of logic, in that all other types of Boolean logic gates (i.e., AND, OR, NOT, XOR, XNOR) can be created from a suitable network of just NAND or just NOR gate(s). They can be built from relays or transistors, or any other technology that can create an inverter and a twoinput AND or OR gate. Hence the NAND and NOR gates are called the universal gates.
For an input of 2 variables, there are 16 possible boolean algebraic functions. These 16 functions are enumerated below, together with their outputs for each combination of inputs variables.
INPUT  A  0  0  1  1 
B  0  1  0  1  
OUTPUT  0  0  0  0  0 
A AND B  0  0  0  1  
A > B *  0  0  1  0  
A  0  0  1  1  
A < B *  0  1  0  0  
B  0  1  0  1  
A XOR B  0  1  1  0  
A OR B  0  1  1  1  
A NOR B  1  0  0  0  
A XNOR B  1  0  0  1  
NOT B  1  0  1  0  
A ≥ B *  1  0  1  1  
NOT A  1  1  0  0  
A ≤ B *  1  1  0  1  
A NAND B  1  1  1  0  
1  1  1  1  1 
The four functions labeled with a "*" are the logical implication functions: "A ≤ B" can be read as "A implies B"; it follows that "A > B" is "A does not imply B". These four functions are less common and are usually not implemented directly as logic gates.
Logic gates are a vital part of many digital circuits, and as such, every kind is available as an IC. For examples, see the 4000 series of CMOS logic chips or the 700 series.
Symbols[]
There are two sets of symbols in common use, both now defined by ANSI/IEEE Std 911984 and its supplement ANSI/IEEE Std 91a1991. The "distinctive shape" set, based on traditional schematics, is used for simple drawings and is quicker to draw by hand. It is sometimes unofficially described as "military", reflecting its origin if not its modern usage. The "rectangular shape" set, based on IEC 6061712, has rectangular outlines for all types of gate, and allows representation of a much wider range of devices than is possible with the traditional symbols. The IEC's system has been adopted by other standards, such as EN 6061712:1999 in Europe and BS EN 6061712:1999 in the United Kingdom.
Type  Distinctive shape  Rectangular shape  Boolean algebra between A & B  Truth table  

AND 
 
OR 
 
NOT 
 
In electronics a NOT gate is more commonly called an inverter. The circle on the symbol is called a bubble, and is generally used in circuit diagrams to indicate an inverted input or output.  
NAND 
 
NOR 
 
In practice, the cheapest gate to manufacture is usually the NAND gate. Additionally, Charles Peirce showed that NAND gates alone (as well as NOR gates alone) can be used to reproduce all the other logic gates.
Symbolically, a NAND gate can also be shown using the OR shape with bubbles on its inputs, and a NOR gate can be shown as an AND gate with bubbles on its inputs. This reflects the equivalency due to De Morgans law, but it also allows a diagram to be read more easily, or a circuit to be mapped onto available physical gates in packages easily, since any circuit node that has bubbles at both ends can be replaced by a simple bubbleless connection and a suitable change of gate. If the NAND is drawn as OR with input bubbles, and a NOR as AND with input bubbles, this gate substitution occurs automatically in the diagram (effectively, bubbles "cancel"). This is commonly seen in real logic diagrams  thus the reader must not get into the habit of associating the shapes exclusively as OR or AND shapes, but also take into account the bubbles at both inputs and outputs in order to determine the "true" logic function indicated. Two more gates are the exclusiveOR or XOR function and its inverse, exclusiveNOR or XNOR. The two input ExclusiveOR is true only when the two input values are different, false if they are equal, regardless of the value. If there are more than two inputs, the gate generates a true at its output if the number of trues at its input is odd ([1]). In practice, these gates are built from combinations of simpler logic gates.  
XOR 
 
XNOR 

List of Logic Gate ICs[]
The 7400 series has several ICs that contain just fourinput logic gates:
IC No.  Gate 

7400  NAND 
7402  NOR 
7404  NOT 
7408  AND 
7432  OR 
7486  EXOR 
DeMorgan equivalent symbols[]
By use of De Morgan's theorem, an AND gate can be turned into an OR gate by inverting the sense of the logic at its inputs and outputs. This leads to a separate set of symbols with inverted inputs and the opposite core symbol. These symbols can make circuit diagrams for circuits using active low signals much clearer and help to show accidental connection of an active high output to an active low input or viceversa.
Storage of bits 8 in a byte[]
Related to the concept of logic gates (and also built from them) is the idea of storing a bit of information. The gates discussed up to here cannot store a value: when the inputs change, the outputs immediately react. It is possible to make a storage element either through a capacitor (which stores charge due to its physical properties) or by feedback. Connecting the output of a gate to the input causes it to be put through the logic again, and choosing the feedback correctly allows it to be preserved or modified through the use of other inputs. A set of gates arranged in this fashion is known as a "latch", and more complicated designs that utilise clocks (signals that oscillate with a known period) and change only on the rising edge are called edgetriggered "flipflops". The combination of multiple flipflops in parallel, to store a multiplebit value, is known as a register. When using any of these gate setups the overall system has memory; it is then called a sequential system since its output can be influenced by its previous state(s).
These registers or capacitorbased circuits are known as computer memory. They vary in performance, based on factors of speed, complexity, and reliability of storage, and many different types of designs are used based on the application.
Threestate logic gates[]
Threestate, or 3state, logic gates have three states of the output: high (H), low (L) and highimpedance (Z). The highimpedance state plays no role in the logic, which remains strictly binary. These devices are used on buses to allow multiple chips to send data. A group of threestates driving a line with a suitable control circuit is basically equivalent to a multiplexer, which may be physically distributed over separate devices or plugin cards.
'Tristate', a widelyused synonym of 'threestate', is a trademark of the National Semiconductor Corporation.
Miscellaneous[]
Logic circuits include such devices as multiplexers, registers, arithmetic logic unitarithmetic units (ALUs), and computer memory, all the way up through complete microprocessors which can contain more than a 100 million gates. In practice, the gates are made from field effect transistors (FETs), particularly metaloxidesemiconductor FETs (MOSFETs).
In reversible logic, Toffoli gates are used.
History and development[]
The earliest logic gates were made mechanically. Charles Babbage, around 1837, devised the Analytical Engine. His logic gates relied on mechanical gearing to perform operations. Electromagnetic relays were later used for logic gates. In 1891, Almon Strowger patented a device containing a logic gate switch circuit (Template:US patent). Strowger's patent was not in widespread use until the 1920s. Starting in 1898, Nikola Tesla filed for patents of devices containing logic gate circuits (see List of Tesla patents). Eventually, vacuum tubes replaced relays for logic operations. Lee De Forest's modification, in 1907, of the Fleming valve can be used as AND logic gate. Ludwig Wittgenstein introduced a version of the 16row truth table in #Logic gates and hardware above, as proposition 5.101 of Tractatus LogicoPhilosophicus (1921). Claude E. Shannon introduced the use of Boolean algebra in the analysis and design of switching circuits in 1937. Walther Bothe, inventor of the coincidence circuit, got part of the 1954 Nobel Prize in physics, for the first modern electronic AND gate in 1924. Active research is taking place in molecular logic gates.
References[]
 Symbols for logic gates. Twenty First Century Books, Breckenridge, CO.
 Tesla's invention of the AND logic gate. Twenty First Century Books, Breckenridge, CO.
 Wireless Remote Control and the Electronic Computer Logic logic elements
 "LEGO Logic Gates". goldfish.org.uk, 2005.
Further reading[]
 Awschalom, D., D. Loss, and N. Samarth, Semiconductor Spintronics and Quantum Computation (2002), SpringerVerlag, Berlin, Germany.
 Bostock, Geoff, Programmable Logic Devices. Technology and Applications (1988), McGrawHill, New York, NY.
 Brown, Stephen D. et al., FieldProgrammable Gate Arrays (1992), Kluwer Academic Publishers, Boston, MA.