China
Africa
Explore chapters and articles related to this topic
Introduction
Published in Wen-Long Chin, Principles of Verilog Digital Design, 2022
Sometimes, digital circuit design also refers to the logic design. A digital function can be generally expressed by the Boolean equation. Boolean algebra is an algebra that deals with binary variables and logic operations. For example, the Boolean equation of the output Y is written as Y=A·B·C¯+A¯·B·C¯+C
Introduction
Published in Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski, Computer Arithmetics for Nanoelectronics, 2018
Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski
Boolean functions are particular functions that can be described in terms of expressions over Boolean algebra, called Boolean formulas. A Boolean formula of n variables is a string of symbols of x1,x2,…,xn, the binary operations of Boolean sum (∨), Boolean product (⋅), unary operation of the complement (-), and brackets (). A Boolean formula is the Boolean function after specification of values, given assignments of variables.
Digital Design—Combinational Logic
Published in Bogdan M. Wilamowski, J. David Irwin, Fundamentals of Industrial Electronics, 2018
Buren Earl Wells, Sin Ming Loo
It can be shown that any combinational function can be implemented using a combination of the basic AND, OR, and NOT operators. The corresponding set of gates is easily implemented in digital hardware in a manner that is dependent upon the underlying device technology. In Boolean algebra, a truth table can be used to show the values of the output for all possible values of the set of inputs. An example of the truth tables for all three of these main types of gates is shown in Figure 20.2. An AND gate is characterized by the fact that all inputs must be at a logic 1 before the output is a 1. An OR gate is characterized by the fact that if any of the inputs are at a logic 1, then the output is also at a logic 1, and a NOT gate simply inverts (switches) the value of its logic input. The functionality of applying the logic operations on the set of inputs to produce an output can also be expressed using Boolean algebra notation. In this notation, the input and outputs of a logical network are represented as variables or constants in much the same way as conventional algebraic statements. The operations performed by these three gates are expressed using standard algebraic symbols, where the AND operation represents Boolean multiplication, and the OR operation represents Boolean addition. By definition, the NOT operation is a unary operator. It is expressed by putting a line over the variable of interest.
Subordination Tarski algebras
Published in Journal of Applied Non-Classical Logics, 2019
In a Boolean algebra, a subordination relation is a relation ≺ satisfying the conditions (S1) to (S4) and the additional condition (Bezhanishvili et al., 2016). So, a pair , where A is a Boolean algebra and ≺ is a subordination is called a subordination Boolean algebra. By the results of Bezhanishvili et al. (2016), Celani (2016), the class of subordination Boolean algebras is equivalent to the class of quasi-modal algebras. We note that if ≺ is a subordination defined in a Boolean algebra A satisfying the conditions (S1) to (S4) and , then for each , as , we have that . Thus, in a Boolean algebra any subordination ≺ satisfies the condition (S5). So, the definition of subordination relation given in Definition 3.2 is equivalent one to given in Bezhanishvili et al. (2016) when A is a Boolean algebra.
MVW-rigs and product MV-algebras
Published in Journal of Applied Non-Classical Logics, 2018
Alejandro Estrada, Yuri A. Poveda
A rig is a structure such that and are commutative monoids, and the product distributes over the sum (Schanuel, 1991 or Castiglioni, Menni, & Botero, 2016, Definition 2.3). In this context, a weak rig is a structure where is a commutative monoid and is a semigroup such that , . Thus both MVW-rigs and rigs are weak rigs. The class of Boolean algebras is an example. Every Boolean algebra is an MV-algebra if we consider the sum defined by the supremum and the negation by the complement. Also if we consider the product as the infimum, every Boolean algebra is an MVW-rig, an MV-rig (Castiglioni et al., 2016) and a (weak) rig.
Fragments of quasi-Nelson: residuation
Published in Journal of Applied Non-Classical Logics, 2023
Let be a -semilattice. The following are equivalent:. is a Boolean algebra.