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Boolean Algebra
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
A Karnaugh map is a diagrammatic representation of a Boolean expression in disjunctive normal form. It consists of a rectangle divided into subrectangles referred to as cells where each cell may be taken to represent a minterm. For a given number of variables the cells within the Karnaugh map represent all the possible minterms which may appear in the disjunctive normal form of a Boolean expression. The minterms are allocated to the cells in such a way that adjacent cells represent minterms in which all the literals are identical except for one which is complemented in one cell but not in an adjacent one.
Boolean Logic
Published in Douglas O. J. deSá, Instrumentation Fundamentals for Process Control, 2019
A truth table is a tabulated list of values of a Boolean expression for each possible combination of the variables in the expression. The writing of a truth table can be reduced to a systematic method of construction that can be applied to any number of Boolean variables. To completely specify the condition of a single variable A, we can write the truth table shown in Table 24.6. To completely specify all the conditions of two variables A and B, we can write the truth table shown in Table 24.7.
Boolean Algebra and Logic Gates
Published in Dale Patrick, Stephen Fardo, Vigyan ‘Vigs’ Chandra, Electronic Digital System Fundamentals, 2020
Dale Patrick, Stephen Fardo, Vigyan ‘Vigs’ Chandra
One important application of Boolean algebra is to express how logic functions are achieved mathematically. A Boolean expression is used to show how an equation responds with respect to its input and output. The inputs and outputs are identified by designated letters. A Boolean expression can be described by a mathematical statement, a logic circuit, or a truth table.
Ideas for Improving the Digital Design Lab
Published in IETE Journal of Education, 2023
Given that the students carry out only about eight experiments, a programming language can be used to perform the simulations of a digital circuit. If the students are well-versed with a programming language such as C or Python [8], it is not difficult for them to write a program to simulate the behavior of a circuit that implements a Boolean expression. For example, consider a magnitude comparator which compares two 4-bit numbers and generates three outputs, equal, greater, and lesser. The logic for generating the three outputs is fairly straightforward. But writing a behavioral description using integers to represent the 4-bit numbers and using comparison operators ==, < and > to compare the numbers is not educational. It is not easy for a teacher to check if the student has resorted to “behavioral description” in the experiment while the expectation was to describe and simulate the underlying logic. The two inputs can be represented by integer arrays. Figure 1 shows the C description of a 4-bit magnitude comparator. The student will have to write another program to generate test cases to test the description. Figure 2 shows a simple C program to generate every pair of two four-bit numbers (there will be 216×216 pairs), automatically create a command of the following form and run the command.
An Online Course Content for Undergraduate Students on Full-Custom Design of a Digital VLSI Circuit Using Open-Source Software
Published in IETE Journal of Education, 2023
The course content and online teaching methodology were applied to the Winter Semester students of 2019–2020 and 2020–2021 during the COVID-19 time. The total strength of the class was 100 and 57, in the 2019–2020 and 2020–2021 Winter Semesters, respectively. The goal of the course evaluation is to test the students’ knowledge and invite their feedback; therefore, the survey model discussed in Ref. [19] was adopted, after showing some more examples based on the concepts presented in this paper. The student learning and knowledge were accessed by giving two questions in the class. Test one consists of tough question 1, and easy question 2, the students were asked to choose one question and asked to draw the stick diagram, and layout it on paper, then simulate using the Microwind-2.6 tool. The tough question has five inputs in the Boolean expression, whereas the easy question has three inputs in the Boolean expression. The test duration was of one hour, after the test, students were asked to send their answers in a pdf format through email. The student documents were corrected, and Table 1 shows the students’ results for Test one according to the questions.
Measuring and enhancing the connectivity reliability of a rail transit network
Published in Transportmetrica A: Transport Science, 2022
Jie Liu, Paul M. Schonfeld, Shuguang Zhan, Qiyuan Peng, Yuhong Liu
Step 2.2: Branch the selected link. The selected link is branched according to Shannon’s decomposition theorem (Bryant 1986) which is shown as: where is the Boolean expression for all variables (links). and represent the connected state ‘1’ and unconnected state ‘0’ of the link , respectively. and represent the Boolean expression when link is connected or not, respectively. The links are branched according to Shannon’s decomposition theorem until the branch determines the state of a terminal node (‘0’ or ‘1’).