File Name: postulates and theorems of boolean algebra .zip
Boolean Algebra is a form of mathematical algebra that is used in digital logic in digital electronics. Albebra consists of symbolic representation of a statement generally mathematical statements.
- Formal Verification of Hardware Components in Critical Systems
- Digital Circuits - Boolean Algebra
- Boolean Algebra Laws and Theorems
Boolean theorems and laws are used to simplify the various logical expressions. In a digital designing problem, a unique logical expression is evolved from the truth table.
Formal Verification of Hardware Components in Critical Systems
Boolean algebraic theorems are the theorems that are used to change the form of a boolean expression. Sometimes these theorems are used to minimize the terms of the expression and sometimes they are used just to transfer the expression from one form to another. Thus, the complement of the product of variables is equal to the sum of their individual complements. Thus, the complement of the sum of variables is equal to the product of their individual complements. Redundancy Theorem : This theorem is used to eliminate the redundant terms. A variable is associated with some variable and its complement is associated with some other variable and the next term is formed by the left over variables, then the term becomes redundant.
Hardware components, such as memory and arithmetic units, are integral part of every computer-controlled system, for example, Unmanned Aerial Vehicles UAVs. The fundamental requirement of these hardware components is that they must behave as desired; otherwise, the whole system built upon them may fail. To determine whether or not a component is behaving adequately, the desired behaviour of the component is often specified in the Boolean algebra. Boolean algebra is one of the most widely used mathematical tools to analyse hardware components represented at gate level using Boolean functions. To ensure reliable computer-controlled system design, simulation and testing methods are commonly used to detect faults; however, such methods do not ensure absence of faults. In this paper, we define a lightweight mathematical framework in computer-based theorem prover Coq for describing and reasoning about Boolean algebra and hardware components logic circuits modelled as Boolean functions. To demonstrate the usefulness of the framework, we 1 define and prove the correctness of principle of duality mechanically using a computer tool and all basic theorems of Boolean algebra, 2 formally define the algebraic manipulation step-by-step procedure of proving functional equivalence of functions used in Boolean function simplification, and 3 verify functional correctness and reliability properties of two hardware components.
Digital Circuits - Boolean Algebra
Hence, it is also called as Binary Algebra or logical Algebra. A mathematician, named George Boole had developed this algebra in The variables used in this algebra are also called as Boolean variables. In this section, let us discuss about the Boolean postulates and basic laws that are used in Boolean algebra. These are useful in minimizing Boolean functions. Either the Boolean variable or complement of it is known as literal.
The postulates of a mathematical system are based on the basic assumptions, which make possible to deduce the rules, theorems, and properties of the system.
Boolean Algebra Laws and Theorems
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In abstract algebra , a Boolean algebra or Boolean lattice is a complemented distributive lattice.