** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.

** Parity function, a Boolean function whose value is 1 if the input vector has an odd number of ones

** The application of Boolean Algebra to relay circuit design was formalized by Claude Shannon in A Symbolic Analysis of Relay and Switching Circuits

** free Boolean algebra

** Boolean algebra ( structure )

** Many embedded applications have a limited amount of physical space for circuitry ; keeping peripherals on-chip will reduce the space required for the circuit board.

** Electronic filter, an electronic circuit which processes signals, for example to remove unwanted frequency components

** Electronic keyboard, a keyboard instrument in which key presses complete an electric circuit

** In circuit analysis and signal processing to represent natural frequency, related to frequency f by ω = 2πf

** Digital trip computer / warning system added to dashboard, ignition circuit monitor system added.

** Temperature sensors for ignition circuit monitor system moved from # 4 and # 8 cylinders to # 3 and # 7 cylinders to improve their efficiency.

** Controlling a high-voltage circuit with a low-voltage signal, as in some types of modems or audio amplifiers,

** Controlling a high-current circuit with a low-current signal, as in the starter solenoid of an automobile,

** A massive short circuit in Seasat's electrical system ends the satellite's scientific mission.

** Circuit analysis, the process of finding the voltages across, and the currents through, every component in an electrical circuit

** Electronic circuit, active electronic components connected in a circuit

** Analog circuit or Digital circuit

** Integrated circuit

** Asynchronous circuit or Synchronous circuit

** Printed circuit board ( PCB )

** Telecommunication circuit

** Balanced circuit

** LC circuit

** Integer circuit, a mathematical object of computational complexity

** Chip bonding, method of wiring some chips ( also from different manufactures ) together on die an integrated circuit

Claude E. Shannon developed much of modern information theory and discovered the application of Boolean logic to digital circuit design theory.

A majority gate is a logical gate used in circuit complexity and other applications of Boolean circuits.

Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit ( which can be calculated from the length of the input ) with polylogarithmic depth and a polynomial number of gates.

Hence Boolean algebra became the foundation of practical digital circuit design ; and Boole, via Shannon and Shestakov, provided the theoretical grounding for the Digital Age.

This system was not put into use until a graduate student from Massachusetts Institute of Technology by the name Claude Shannon noticed that the Boolean algebra he learned was similar to an electric circuit.

In computer science, circuit diagrams are especially useful when visualizing different expressions using Boolean Algebra.

The most common complexity class involving advice is P / poly where advice length f ( n ) can be any polynomial in n. P / poly is equal to the class of decision problems such that, for every n, there exists a polynomial size Boolean circuit correctly deciding the problem on all inputs of length n. One direction of the equivalence is easy to see.

If, for every n, there is a polynomial size Boolean circuit A ( n ) deciding the problem, we can use a Turing machine that interprets the advice string as a description of the circuit.

Consider the circuit minimization problem: given a circuit A computing a Boolean function f and a number n, determine if there is a circuit with at most n gates that computes the same function f. An alternating Turing machine, with one alternation, starting in an existential state, can solve this problem in polynomial time ( by guessing a circuit B with at most n gates, then switching to a universal state, guessing an input, and checking that the output of B on that input matches the output of A on that input ).

The usual purpose of doing Boolean algebra is to simplify the design of a digital circuit that performs a function, either to minimize the number of gates, or to minimize the time for the value of the function to settle down after a change in its input ( s ), or some other practical criterion.