# What is xy + xy Boolean algebra

## Boolean Algebra (1)

### Information processing at the bit level Information processing at bit level Dr. Christian Herta November 5, 2005 Introduction to Computer Science - Information Processing at Bit Level Dr. Christian Herta basics of information processing

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### Basics of information processing Basics of information processing Information is represented in binary form in the computer. The binary represented data should be processed in the computer, i.e. computer circuits must exist,

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### Switching function, definition Switching function, definition Let S = {0,1}. Then a mapping f: S n S is called a switching function. = f (x n-1, x n-2, ... ,,,), x n-1, x n-2, ... ,,, S xi X = (x n-1, x n- 2, ... ,,,) input variable input vector

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### A.1 Switching functions and switching networks Switching functions and switching networks A. Switching functions and switching networks 22 Prof. Dr. Rainer Manthey Informatik II Significance of the binary system for the computer structure Since the beginning of the development of computer hardware

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### 3 Boolean algebra and propositional logic 3 Boolean Algebra and Propositional Logic 3- Boolean Algebra Formal Foundations of Computer Science I Fall Semester 22 Robert Marti Lecture partly based on documents from Prof. emer. Helmut Schauer

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### 2. Switching functions and their representation 2. Switching functions and their representation x y z Switching algebra Switching circuits and terms Switching functions Duality principle Boolean algebra Representation of switching functions 60 Switching algebra We investigate

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### 2. Switching functions and their representation 2. Switching functions and their representation x y z Switching algebra Switching circuits and terms Switching functions Duality principle Boolean algebra Representation of switching functions 58 Switching algebra We investigate

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### Technical Informatics - An Introduction Martin Luther University Halle-Wittenberg Department of Mathematics and Computer Science Chair for Computer Engineering Prof. P. Molitor Computer Engineering - An Introduction Boolean Functions - Basics

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### Introduction to Computer Science I Introduction to Computer Science I Arithmetic and Bitwise Operators in the Binary System Prof. Dr. Nikolaus Wulff Operations with binary numbers When calculating with binary numbers, there are the completely normal arithmetic

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### Introduction to Boolean Algebra Introduction to Boolean Algebra Introduction to Boolean Algebra 1 Binary quantity A quantity (a variable) that can have exactly 2 values ​​mathematically: false statement true statement technically: switched off

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### II. Basics of programming II. Basics of programming II.1. Number systems and elementary logic 1.1. Number stems 1.1.1. Whole numbers Whole numbers are shown in the decimal as a sequence of digits 0, 1, ..., 9, e.g. 123

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### Part II. Switching functions Part II Switching functions 1 Part II.1 Number representation 2 b-adic systems Let b IN with b> 1 and E b = {0, 1, ..., b 1} (alphabet). Then every fixed point number z (with n places in front of the decimal point and k places after the decimal point)

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### Part 1: Digital Logic Part 1: Digital logic Contents: Boolean algebra, combinatorial logic, sequential logic, short digression, technological basics, programmable logic modules. 1 Analog and digital hardware at

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### Technical informatics I Computer structures Dario Linsky winter semester 200/20 Part 2: Basics of digital circuits Overview Logical functions and gates Transistors as electronic switches Integrated circuits

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### Algorithms & programming. logic Algorithms & Programming Logic Propositional Logic Subject of the investigation Links between statements are investigated. Statements What a statement is is not considered, but every statement is

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### Switching algebra and combinational logic Switching algebra and combinational logic. Digital electrical circuits 2. Description by logical expressions 3. Boolean algebra 4. Switching functions 5. Synthesis of circuits 6. Switching networks * Die

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### 2. Digital technology and design process tutorial 2. Tutorial Digital Technology and Design Process Tutorial No. 9 Alexis Tobias Bernhard Faculty of Computer Science, KIT University of the State of Baden-Württemberg and national research center in the Helmholtz Association

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### Part III. Switching networks and their optimization Part III Switching networks and their optimization 1 Part III.1 Switching networks 2 Example 1 Switching network for xor with {+ ,,} xyxy 0 0 0 0 1 1 1 0 1 1 1 0 DNF: xy = xy + xy 3 Example 2 xor using nand -shortcut;

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### Embedded Systems Introduction to Embedded Systems Lecture 7 Bernd Finkbeiner 03/12/2014 [email protected] Prof. Bernd Finkbeiner, Ph.D. [email protected] 1 switching functions! Switching function:

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### Computer Science A (Author: Max Willert) 2. Exercise sheet winter semester 2012/2013 - sample solution computer science A (Author: Max Willert) 1. Logic in everyday life (a) Restaurant A advertises with the slogan Good food is not cheap !, the restaurant next to it

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### Computer structures WS 2012/13 Computer structures WS 202/3 Boolean functions and switching networks Representations of Boolean functions (repetition) Normal forms of Boolean functions (repetition) Representation of Boolean functions

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### 5th lecture: normal forms 5th lecture: Normal forms Repetition Complete systems Minterme Maxterme Disjunctive normal form (DNF) Conjunctive normal form (CNF) 1 XOR (antivalence) X X X X X X (X X) (X X) 1 2 1 2 1 2 1 2 1

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### 5. Propositional logic and switching algebra 5. Propositional logic and switching algebra Propositional forms and propositional logic Boolean terms and Boolean functions Boolean algebra Switching algebra Switching networks and switching mechanisms R. The 1 statements information often

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### What happened so far: classic propositional logic What happened so far: classic propositional logic classic propositional logic: syntax, semantics equivalence between formulas ϕ ψ iff. Mod (ϕ) = Mod (ψ) important equivalences, e.g. Double Negation Elimination, DeMorgan Laws,

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### Information display Information display Signals and logic Basics of Boolean algebra Signals and logic (2) Basics d. Information Theory [Logarithm Repetitorium] Number systems and their application Signals and

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### Conjunctive and disjunctive normal forms Conjunctive and disjunctive normal forms After it was discussed how to interpret Boolean terms with an assignment of the variables and that every Boolean term represents a Boolean function,

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### Logic (Teschl / Teschl 1.1 and 1.3) Logic (Teschl / Teschl 1.1 and 1.3) A statement is a proposition from which one can clearly decide whether it is true (true, = 1) or false (false, = 0). Examples a: 1 + 1 = 2 b: Darmstadt is in Bavaria.

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### General statements Generally valid statements Definition 19 A (propositional) formula p is called generally valid (or a tautology) if p is true under every assignment. A (propositional) formula p is called

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### Logic for computer scientists Lecture Logic for Computer Scientists 5. Propositional Logic Normal Forms Bernhard Beckert University of Koblenz-Landau Summer Semester 2006 Logic for Computer Scientists, SS 06 p.1 Normal Forms Definition: Literal Atom (propositional logic

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### Signal processing 1 TiEl-F000 summer semester 2008 Signal processing 1 (lecture number 260215) 2003-10-10-0000 TiEl-F035 Digital technology 2.1 Logic levels in digital technology In binary circuits, two represent defined

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### Discrete Mathematics Basics Basics of Discrete Mathematics Prof. Dr. Romana Piat WS 25/6 General information Lectures: ./ C Train D (Wed., 3rd block + Thu., 4th block, y-raster) Train E (Tue., 5th block + Thu.,. Block, y-grid)

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### System-oriented computer science 1 System-oriented computer science. Basics of digital circuits. 8 Switching networks from gates and lines. 9 Boolean algebra. Minimization of Boolean functions. CMOS Komplegatter The next function,

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### Basics of computer architecture Basics of Computer Architecture [CS3100.010] Winter Semester 2014/15 Heiko Falk Institute for Embedded Systems / Real-Time Systems Engineering and Computer Science Ulm University Chapter 2 Combinatorial

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### Mathematical Foundations I Logic and Algebra Logic and Algebra Dr. Tim Haga October 21, 2016 1 Propositional Logic First Terms Logical Operators Disjunctive and Conjunctive Normal Forms Logical Inference Dr. Tim Haga 1/21 Preliminaries Last

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### Basics of digital technology Basics of digital technology A systematic introduction by Prof. Dipl.-Ing. Erich Leonhardt 3rd, edited edition With 326 pictures, 128 tables, numerous examples and exercises with solutions

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### Chapter 3: Boolean Algebra Contents: 3.1 Basic operations and laws 3.2 Boolean functions and their normal forms 3.3 Simplifying Boolean expressions 3.4 Logical circuits 3.1 Basic operations and laws

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### 6th lecture: Minimal forms 6th Lecture: Minimal Forms Repetition Minterme Maxterme Disjunctive Normal Form (DN) Conjunctive Normal Form (KN) Minimal Form KV-Diagrams 24..26 is canceled due to lecturer excursion 2 normal forms

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### 2 switching algebra or Boolean algebra * 9 2 Switching algebra or Boolean algebra * The two-valued logic is of particular importance in computer development, since data with physical quantities is represented particularly well by two values

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### TU9 propositional logic. Daniela Andrade TU9 Propositional Logic Daniela Andrade daniel[email protected] 18.12.2017 1/21 Small note My slides are based on Carlos Camino's DS Trainer, which you can find at www.carlos-camino.de/ds;) 2 /

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### Introduction to Computer Science Introduction to Computer Science Lecture given by Prof. Dr. rer. nat. E. Bertsch Script written by Sebastian Ritz December 7th, 2005 1 Table of contents 1 What is meant by computer science 3 2 Structure

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### Algebraic structures and associations CHAPTER 4 Algebraic Structures and Associations Definition 4.1. Let M be a set. A figure: M M M is called a (two-digit) connection in M. One also writes a b: = (a, b) with a, b M.

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### 6. Boolean algebras 6. Boolean algebras 6.1 Definitions A Boolean algebra is an algebra S ,,,, 0, 1 ,, are binary, is a unary operator, 0 and 1 are constants. The following applies: 1 and are associative and commutative.

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### 2. Lecture: Boolean Algebra 2nd lecture: Boolean algebra repetition Coding, decoding Boolean algebra AND-, OR-operation, negation Boolean postulates Boolean laws 1 repetition 2 bits and bit sequences Bit: Unit of measurement

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### Boolean functions and circuits Boolean functions and circuits The OR function (disjunction) and the AND function (conjunction), xy 0 0 0 0 1 1 1 0 1 1 1 1 xy 0 0 0 0 1 0 1 0 0 1 1 1 1 (implication function) , (reverse

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### Representation of negative binary numbers Representation of negative binary numbers Observation for any binary number B, e.g. B = 110010: B + NOT (B) ---------------------------------------- ------ = B + NOT (B) 1 + (Carry) -------------------------------- --------------

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### Switching algebra. Prof. Metzler Switching algebra 1 Switching algebra (oolsche lgebra) George Oole, British mathematician, 1815-1864 "The mathematical analysis of logic" 1847, 1854 1938 directs

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### N bit binary numbers (signed) N bit binary numbers (signed) n bit representation is a window on the first n digits of the binary number 00000000000000000000000000000000000000000000000110 = 6 11111111111111111111111111111111111111111111111101

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### Introduction to theoretical computer science Introduction to Theoretical Computer Science Week 4 Harald Zankl Institute for Computer Science @ UIBK Winter Semester 2014/2015 Summary Summary of the last course Modus Ponens A B B A MP Axioms for

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### 183.580, WS2012 Exercise groups: Mon., 22.10. VU Fundamentals of Digital Systems Exercise 2: Numerics, Boolean Algebra 183.580, WS2012 Exercise groups: Mon., 22.10. Exercise 1: Binary floating point arithmetic addition & subtraction The numbers are given: A

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### 1 Algebraic Structures Prof. Dr. Rolf Socher, FB Technik 1 1 Algebraic Structures In mathematics one is often concerned with sets on which certain operations are defined. It often happens that these operations

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