# Boolean logic

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(mathematics)

## Boolean logic

The "mathematics of logic," developed by English mathematician George Boole in the mid-19th century. Its rules govern logical functions (true/false) and are the foundation of all electronic circuits in the computer. As add, subtract, multiply and divide are the primary operations of arithmetic, AND, OR and NOT are the primary operations of Boolean logic. Boolean logic is turned into logic gates on the chip, and the logic gates make up logic circuits that perform functions such as how to add two numbers together.

Various permutations of AND, OR and NOT are used, including NAND, NOR, XOR and XNOR. The rules, or truth tables, for AND, OR and NOT follow. See Boolean search, binary, logic gate and Bebop to the Boolean Boogie.

An AND Gate (Wired in Series) AND requires both inputs to be present in order to provide output. When both inputs pulse both switches closed, current flows from the source to the output.

Wired in patterns of Boolean logic and in less space than a postage stamp, transistors in one of today's high-speed chips collectively open and close quadrillions of times every second. If you are curious about how it really works down deep in the layers of the silicon, read the rest of "Boolean logic," then "chip" and, finally, "transistor." It is a fascinating venture into a microscopic world.

The following AND, OR and NOT examples use mechanical switches to show open and closed transistors. The switching part of an actual transistor is solid state (see transistor).

An AND Gate (Wired in Series) AND requires both inputs to be present in order to provide output. When both inputs pulse both switches closed, current flows from the source to the output.

An OR Gate (Wired in Parallel) OR requires only one of the two inputs to be present in order for current to flow from the source to the output.

A NOT Gate (Input Is Reversed) No pulse in puts current out (as shown). A pulse in puts no current out, as follows: an input pulse closes switch #1 and the current goes to #2. Switch #2 is normally closed, and a pulse from #1 opens it and stops the flow.

The Hierarchy The gates make up circuits, and circuits make up logical devices, such as a CPU. We're going to look at a circuit that is present in every computer. It adds one bit to another.

Adding Two Bits Together The half-adder circuit adds one bit to another and yields a one-bit result with one carry bit. This circuit in combination with a shift register, which moves over to the next bit, is how a string of binary numbers are added. This diagram shows the four possible binary additions for two bits.

The Half-Adder Circuit Trace the current through the example above. See how AND, OR and NOT react to their inputs. The 1 is represented in red (flow of current), and the 0 in blue (no current). Try it yourself below.

Try It Yourself Print this diagram and try your Boolean skill. Review the combinations of 0 and 1 above and pick any pair. With a pen or pencil, draw a line to represent a 1. Draw nothing for 0, and see if you can get the right answer.

Try It Yourself Print this diagram and try your Boolean skill. Review the combinations of 0 and 1 above and pick any pair. With a pen or pencil, draw a line to represent a 1. Draw nothing for 0, and see if you can get the right answer.
References in periodicals archive ?
For this sake, we work on the expression tree of the formula, looking for its transformation into an equivalent family of boolean circuits of minimum depth.
Key words: Computation Theory, Relational Databases, Boolean Circuits, Query
86]), we show how a query to a given relational database, expressed through a formula of First Order Logic (FO), can be translated into a finite subfamily of boolean circuits in the complexity class AC.
Given an r-ary query and a database with domain of size n, a finite subfamily of boolean circuits C = {[C.
This problem considered in the context of the Turing machine model, consists of a machine whose input is a formula and a natural number n, and whose output is a subfamily of boolean circuits C, as we described previously.
We only consider in this work boolean circuits with gates with either one or two inputs (bounded fan-in) and whose output is used as input in at most one other gate (bounded fan-out).
Finally, we build the finite subfamilies of the corresponding boolean circuits for a given database, and we give a strategy for the minimization of the depth of the circuits in the subfamily, which is applicable in some cases.
Our boolean circuits belong to the complexity class NC, which makes formal the notion of well parallelizability of a function.
1992] and Papadimitriou and Yannakakis [1985], applied to results in Stewart [1991], and the simulation of Boolean circuits by a [DATALOG.
The query program involves a subprogram that simulates Boolean circuits by which the relations in the input structure encoding the graph can be computed.

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