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# A Study on Complemented Lattices and Boolean Function

 dc.contributor.advisor Rahman, Prof. Dr. Md. Bazlar dc.contributor.author Das, Kishore Kumar dc.date.accessioned 2018-08-13T19:47:50Z dc.date.available 2018-08-13T19:47:50Z dc.date.copyright 2010 dc.date.issued 2010-06 dc.identifier.other ID 0651503 dc.identifier.uri http://hdl.handle.net/20.500.12228/372 dc.description This thesis is submitted to the Department of Mathematics, Khulna University of Engineering & Technology in partial fulfillment of the requirements for the degree of Master of Philosophy in Mathematics, June 2010. en_US dc.description Cataloged from PDF Version of Thesis. dc.description Includes bibliographical references (pages 63-63). dc.description.abstract In this thesis the nature of Complemented lattice and Boolean function is studied. Lattice theory is a part of Mathematics. In Modern algebra, Abstract algebra and Boolean function are Lattice theory play important role. A non empty set P together with a binary relation R is said to form a partially ordered set or a posel if the following conditions hold: en_US Reflexivity Anti-symmetry Transitivity A poset (L≤) is said to form a lattice if for every a, b ϵ L if Sup {a,b} and Inf { a,b}exist in L. A lattice is said to be complemented lattice if every element has complement. In this thesis we give several results on complemented lattice, Boolean function and Boolean algebra which will certainly extend and generalize many results in lattice theory. The thesis contains four chapters. Chapter one: We have discussed the basic definition of set, partially order set, relations, functions etc. Chapter two: We have discussed lattice, sublattice, convex sublattice, complemented lattice, ideal, Filter, Prime ideal, Principle ideal and Principle Dual ideal. We have proved that two bounded lattices A and B are complemented if and only if A x B is complemented. In this chapter we have also discussed the definition of upper bound, least upper bound, lower bound, greatest lower bound and relatively complemented lattice, and established relation among them. We also studied some other properties of these concepts and we have showed that two lattices A and B are relatively complemented if and only if the cross product of two lattices A and B is relatively complemented. Chapter three: We have discussed Boolean algebra, Boolean lattice and Boolean function. Let (A, ˄, ˅, ', 0, 1) be a Boolean algebra. Expressions involving members of A and the operations ˄, ˅ and complementation are called Boolean expressions or Boolean polynomials. For example, x ˅ y', x, x ˄ 0 are etc. all Boolean expressions. Any function specifying these Boolean expressions is called a Boolean function. Thus if/(x,y) =X ˄ y then f is the Boolean function and x ˄ y is the Boolean expressions (or value of the function f). Since it is normally the functional value (and not the function) that we are interested in, we call these expressions the Boolean function. We will denote least and greatest elements of a Boolean algebra by 0 and 1 respectively. In fact, most of the times we will confine ourselves to Boolean algebras that contain only these two elements. We also discus in this chapter Disjunctive Normal form (DN form), Conjunctive Normal form (CN form), Length and Cover. A Boolean function is said to be in DN form in n variables x1, x2, …… xn if it can be written as join of terms of the type f1(x1) ˄ f2 (x2)A ........ ˄ fn (xn ) where fi(xi ) = xi, or x’i for all i =1,2,3,…… ,n and no two terms are same. Also 1 and 0 are said to be in DN form. We also prove them theorem: 'Every Boolean function can be put in DN form'. Here we give several results on DN form, CN form, Homomorphisom, Iso-morphisom and Indomorphisom. Chapter four : In this chapter we have studied series combination, parallel combination, don't care condition and Bridge circuits. By a switch we mean a contact or a device in an electric circuit which lets the current to flow through the circuit. The switch can assume two states 'closed 'or 'open '(ON or OFF). In the first case the current flows and in the second the current does not flow. We will use a,b,c, . . ..... .....x,y,z........etc. to denote switches in a circuit. Two switches a,b are said to be connected 'in series' if the current can pass only when both are in closed state and current does not flow if any one or both are open. Two switches a, b, are said to be connected 'in parallel' if current flows when any one or both are closed and current does not pass when both are open. In this chapter we also solve some circuit problems. dc.description.statementofresponsibility Kishore Kumar Das dc.format.extent 63 pages dc.language.iso en_US en_US dc.publisher Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh. en_US dc.rights Khulna University of Engineering & Technology (KUET) thesis/dissertation/internship reports are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. dc.subject Complemented Lattices en_US dc.subject Boolean Function en_US dc.title A Study on Complemented Lattices and Boolean Function en_US dc.type Thesis en_US dc.description.degree Master of Philosophy in Mathematics dc.contributor.department Department of Mathematics
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