Boolean logic

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(mathematics)

Boolean logic

The "mathematics of logic," developed by English mathematician George Boole in the mid-19th century. Just as add, subtract, multiply and divide are the primary operations of arithmetic, AND, OR and NOT are the primary logical operators of Boolean logic and building blocks of every digital circuit. NAND (not AND), NOR (not OR) and XOR (exclusive OR) are variations of AND, OR and NOT (see NAND, NOR and XOR). See Boolean search, binary, logic gate and Bebop to the Boolean Boogie.

Decimal vs. Binary Decimal arithmetic holds 10 values (0 to 9) in each digit position. In binary, there are only two (0 and 1). Note below the four possible result and carry bits when adding two binary digits (bits) together.

AND, OR and NOT Gates
Transistors wired in series and parallel patterns make up these "gates," which accept inputs of 0 (no pulse) or 1 (a pulse) and generate outputs of 0 or 1. While AND requires that both inputs are 1 to generate an output of 1, OR only needs one input to be 1. NOT reverses the input. For a diagram of these actions, see Boolean gates.

Decimal vs. Binary Decimal arithmetic holds 10 values (0 to 9) in each digit position. In binary, there are only two (0 and 1). Note below the four possible result and carry bits when adding two binary digits (bits) together.

Add a One and Zero Trace the red 1 (pulse, current) and white 0 (no pulse, no current) through the gates and notice their outputs. This half-adder circuit is in every CPU.

Add a One and Zero Trace the red 1 (pulse, current) and white 0 (no pulse, no current) through the gates and notice their outputs. This half-adder circuit is in every CPU.

Trace the Flow Yourself Choose any two binary digits as input and trace them through this half-adder circuit (0 is no current; 1 is current). Watch how the output is generated.

Trace the Flow Yourself Choose any two binary digits as input and trace them through this half-adder circuit (0 is no current; 1 is current). Watch how the output is generated.

Patterns of Boolean Logic Transistors (depicted here as mechanical switches) make up gates. Gates make up circuits, and circuits make up every digital device. There are millions and billions of transistors in modern chips, and they fit in an area smaller than a postage stamp. For more details, see Boolean gates. See transistor and chip.
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References in periodicals archive ?
[1992] and Papadimitriou and Yannakakis [1985], applied to results in Stewart [1991], and the simulation of Boolean circuits by a [DATALOG.sup.[disjunction]] program without [not equal to] and negation under minimal model semantics.
The query program involves a subprogram that simulates Boolean circuits by which the relations in the input structure encoding the graph can be computed.
From [C.sub.I], Boolean circuits [C.sub.P] and [C.sub.N] are constructible in polynomial time that compute P and N, respectively, where each argument x, y, u, and v is encoded by a string of [log n] bits.

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