American Journal of Engineering Research (AJER) 2024 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-13, Issue-6, pp-86-97 www.ajer.org Research Paper Open Access www.ajer.org Page 86 Boolean Algebra and Proving Logic Circuits through Logicly Senad Orhani 1 , Xhevdet Spahiu 2* , Yllka Kusari-Radoniqi 3 1 Department of Education, Faculty of Education, University of Prishtina "Hasan Prishtina", Prishtina 10000, Kosovo 2 Department of Computer Science, Faculty of Computer Science, College of AAB, Prishtina 10000, Kosovo 3 Department of Informatics, Faculty of Natural Sciences and Mathematics, University of Tetovo, 1200 Tetovo, Republic of North Macedonia Corresponding Author: xhevdet.spahiu@universitetiaab.com ABSTRACT : Modern digital computers are built using techniques and symbolism from a field of mathematics called modern algebra. Algebraists have studied for over a hundred years mathematical systems called Boolean Algebra. Nothing could be simpler and more normal for human reasoning than the rules of a Boolean Algebra, because these derive from the studies of how we reason, which reasonings are valid, what constitutes evidence, and other topics. The following design is intended to explain the use of Logicly for validating logic circuits from Boolean Algebra. Logicly is a user interface environment that provides a simple and effective platform for modeling and analyzing these expressions. In this paper, we will explain how we can prove their accuracy. The research results showed that the Logicly program has an excellent performance in validating logic circuits and is an efficient and reliable tool for this purpose. The Logicly program also showed good consistency in the output responses to changes in the inputs of the logic circuits. KEYWORDS: Boolean Algebra, Proving, Logic Circuits, Logicly, Truth table --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 12-06-2024 Date of acceptance: 24-06-2024 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION Algebra (from the Arabic: al-jabr, meaning "reuniting the broken parts" (Boyer 1991) and "setting the bones" (Claudia and Van Oystaeyen 2017) is one of the most important branches of mathematics, along with number theory, geometry, and analysis. In its form more generally, algebra is the study of mathematical symbols and the rules for manipulating these symbols (Herstein 1965). Algebra is generally considered to be essential to any study of mathematics, science, or engineering, as well as applications such as medicine, economics, computer science, and many other disciplines (Whitesitt 2010). The name Boolean Algebra honors a fascinating English mathematician, George Boole (1815-1864), who in 1854 published a classic book entitled "An Inquiry into the Laws of Thought, on which the Mathematical Theories of Logic and Probability are Founded" Boole's stated aim was to perform an analysis of mathematical logic. Beginning with his tracing of the laws of logic, Boole constructed a "logical algebra". Boolean Algebra first introduced problems that had arisen in the design of switching relay circuits in 1938 by Claude E. Shannon. He presented a method of representing any circuit consisting of combinations of switches and staffs by a series of mathematical expressions, and an analysis was devised for the manipulation of these expressions. The calculation used was shown to be based on the rules of Boolean Algebra (Schardijn 2016). Boolean Algebra, like any other axiomatic mathematical structure or algebraic system, can be characterized by specifying a number of basic things (Boole 1854): o The domain of algebra, that is, the set of elements beyond which algebra is defined o A set of operations performed on elements o A set of postulates, or axioms, accepted as premises without proof o A set of consequences called theorems, laws, or rules, are derived from postulates Boolean algebra is defined as a set of two elements {0,1}, on which 3 basic operations can be applied: addition, multiplication, and complement. For these three operations, they use the logical operators "or" (OR),