DeMorgan's Theorem Research Articles

Some Aspects of Neutrosophic Set Theory

This article aims to introduce my work on constructing neutrosophic set theory. Neutrosophy is a new branch of mathematical systems proposed by Smarandache in 1995 as an extension of an intuitionistic fuzzy set according to three-degree membership functions. The new perspective of neutrosophic sets consists of two paths, the first path depends on the three degrees including the degree of truth-membership, degree of indeterminacy-membership, and degree of false-non-membership respectively, while the second path depends on the generated classical set by three types of neutrosophic sets and study the concepts of neutrosophic theory. In previous work, I Presented the concepts of neutrosophic sets including universal, empty, compliment, and subsets. I also explored neutrosophic operations and their properties, such as neutrosophic unions and neutrosophic intersections. In this article, I shall delve into additional materials and theorems related to these concepts and discuss neutrosophic and symmetric differences, including their properties. Furthermore, I shall present a generalization of De Morgan's theorem.

On the Generalization of Neutrosophic Set Operations: Testing Proofs by Examples

This article aims to present more facts about neutrosophic set theory such as classical set theory. It includes; more connection between operations on the neutrosophic set theory of three types such as union, intersection, difference, and their generalization. further, the generalization of union, intersection, and complement with DE-Morgan's theorem. In addition, we studied the Cartesian Products and their properties. This work opens a new parallel path of a new neutrosophic structure of degree of membership functions that does not depend on intuitionistic fuzzy set and fuzzy set respectively.

Process-Quality Evaluation for Wire Bonding With Multiple Gold Wires

Taiwan is a world-leader in wafer foundry services and IC packaging and testing. Wire bonding is a crucial process in the overall IC-packaging industry chain. Thus, this paper proposes a process-quality evaluation model for wire bonding with multiple gold wires. We chose process quality indices as a tool of evaluation fully mirroring process yield and quality levels. These indices contain unknown parameters and thus require sample data to estimate. We first derived the uniformly minimum variance unbiased estimator of the indices and calculated the upper confidence limits of the indices based on DeMorgan's theorem and Boole's inequality. The upper confidence limits of the indices were then employed to create a confidence interval-based fuzzy membership function, in order to improve the accuracy of estimation as well as solve the problem of uncertainty of the measured data. Next, we obtained the fuzzy critical value and used index estimates and the fuzzy critical value to establish fuzzy test rules. Next, we marked the fuzzy critical value on the axes of a radar chart, which is a visualization evaluation tool, and connected neighboring critical points to create a critical region in the form of a regular polygon. The observed values of the indices were then marked on the axes to produce a visualized fuzzy radar evaluation chart. This fuzzy radar evaluation chart has a solid foundation in statistical inference, and evaluation rules were established using precise fuzzy test methods. Not only is this fuzzy radar evaluation chart easy to use, but it also reduces the chance of misinterpretations made by sampling errors, so that the accuracy of evaluation can be enhanced.

Negating a Generalized Cut Sequence: Bridging the Gap Between Dynamic Fault Trees Quantification and Sum of Disjoint Products Methods

Dynamic fault trees (DFTs) are powerful tools to model industrial systems having dynamic failure mechanisms, such as sequence- and function-dependent failure behaviors. Yet for large and complex DFTs, their quantitative analyses are still of great challenges. Up to now, many researchers have presented several approaches to deal with this problem, and among which, the sum of disjoint products (SDP) methods, such as dynamic binary decision tree, sequential binary decision diagram (SBDD), and improved SBDD, have proven to be an efficient way. In SDP methods, negating a generalized cut sequence is an unavoidable task. Yet, for a complex cut sequence expression where normal, cold and warm spares basic events coexist, its negating operation is still difficult and needs to be further studied. In this paper, based on De Morgan theorem, improved explicit formulas for negating a generalized cut sequences are presented. The new concept of universal set of basic event and its operating rules are proposed to deduce the simplified expressions of general enforcing occurring cut sequences and warm spares occurring cut sequences. To validate the presented approaches, a typical system DFT is analyzed. The results indicate the reasonability and effectiveness of the improved negating formulas. Copyright © 2016 John Wiley & Sons, Ltd.

Multi-output majority gate-based design optimization by using evolutionary algorithm

In this paper, a novel efficient method for optimizing multi-output majority gate based designs is proposed. Majority gate is a fundamental Boolean operator in some nano-scale technologies such as quantum-dot cellular automata (QCA). As a result, the design optimization must be directly implemented on majority gates instead of optimizing the design for AND–OR gates. In some other nanotechnologies, a fundamental element is Minority gate which could be simply converted to majority gate by the De Morgan's theorem. Here, the proposed optimization method works on the basis of evolutionary computation and can reduce both the number of majority gates and the worst-case delay of the circuit. The method is compared to some other optimization algorithms and its efficiency is verified.

Use of Demorgan's theorem in system reliability studies

A method of obtaining a minimized system reliability or unreliability expression using Demorgan's theorem is presented. Some formulae based on this theorem are provided, from which the minimized expression of system unreliability can be obtained by knowledge of pathsets. At the same time, this expression can also provide minimum cutsets of the system easily. On the other hand, one can obtain the minimum pathsets of a system without much difficulty, if the minimum cutsets are known.

Multilevel Nand (NOR) Implementation of Logical Functions

A scheme for implementing multilevel NAND (NOR) network of a logical function is presented. The method avoids the need for application of De Morgan's theorem and eliminates the tedium of drawing AND-OR logic diagrams. The analysis of NAND (NOR) logic diagrams to determine the output functions in AND-OR logic is also made easier by the method.

Vector Boolean Algebra and Calculus

In this paper, binary-vector Boolean algebra, a generalization of ordinary Boolean algebra [6], is introduced. In this algebra, every element is represented by a binary vector and in addition to ordinary AND, OR, and NOT operations, a new operation called the rotation operation which rotates (rightward or leftward) the components of a binary vector is introduced. Moreover, the NOT or COMPLEMENTATION operation is extended to a more general operation called the generalized complement which includes the total complement (ordinary complement), the null complement (no complement), and newly introduced partial complements. Because of this generalization, all axioms and theorems of ordinary Boolean algebra are generalized. In particular, it is shown that DeMorgan's theorem, Shannon's theorem, and the expansion theorem are generalized into more general forms which include their corresponding ordinary version as a special case. It is also shown that any multivalued logic truth table can be represented by a vector Boolean function. Three compact canonical (sum-of-products and product-of-sums) forms of this function are presented.

Techniques for the Transformation of Logic Equations

A generalized DeMorgan's theorem is presented that can be applied to Boolean equations with any number of complementations. The known form of the DeMorgan's theorem is a special case of the generalized theorem with one inversion. This new theorem simplifies and generalizes the transformation of Boolean functions to their equivalent sum of product forms.

The Relationship Between Multivalued Switching Algebra and Boolean Algebra Under Different Definitions of Complement

The relationship between multivalued switching algebra and Boolean algebra is presented by introducing different definitions for the complements of multivalued variables. For every definition introduced, the paper points out which Boolean algebra theorems are valid for multivalued cases, which are invalid, and gives proofs to substantiate the claim. It is shown that DeMorgan's theorem holds for all four definitions of complement given in this paper. One definition allows us to map the multivalued variables into binary variables. Under this definition, all axioms and theorems of Boolean algebra are satisfied and can be used for minimization of any multivalued switching function f. Illustrative examples for minimizing f and its complement f are given.

A generalization of Nelson's algorithm for obtaining prime implicants

In the design of multiple input-output logical systems, the authors have previously formulated and proved a computer program for obtaining optimal system representations satisfying three optimality criteria [1]. In that program, the need for obtaining prime implicants occurred. Two algorithms for this purpose are available in the literature; one due to Quine [2] and one due to Nelson [3], [4]. Nelson's, briefly, is to express the function (whose prime implicants are desired) as a product of sums of literals, multiply out, and discard repeated literals save one and proper multiples. (To obtain the function in the initial form, one can start with the complement of the function expressed as a sum of products and complement it using De Morgan's theorem.) In using these algorithms, it became apparent to us that in certain cases one of these two algorithms required less computer time than the other one. In particular, if the given system has a large number of don't cares then the Nelson algorithm seemed to be preferred [5]. Also in the case of a single function, if there are few zeroes in the truth table representation for the function, then Nelson's algorithm is preferable. It also became apparent to us that in certain situations one needs a generalized version of the Nelson algorithm.