Boolean logic. Boolean Algebra. Introduction to Computer Yung-Yu Chuang NOT AND NOT

Size: px
Start display at page:

Download "Boolean logic. Boolean Algebra. Introduction to Computer Yung-Yu Chuang NOT AND NOT"

Transcription

1 oolean lgebra oolean logic ased on symbolic logic, designed by George oole oolean variables take values as or. oolean expressions created from: NOT, ND, OR Introduction to Computer ung-u Chuang with slides by Sedgewick & Wayne (introcs.cs.princeton.edu), Nisan & Schocken ( and Harris & Harris (DDC) 2 NOT ND X X X X Digital gate diagram for NOT: Digital gate diagram for ND: NOT ND 3 4

2 OR Operator Precedence X+ Examples showing the order of operations: NOT > ND > OR Digital it gate diagram for OR: OR Use parentheses to avoid ambiguity 5 6 Defining a function Defining a function Description: square of x minus lgebraic form : x 2 - Enumeration: x f(x) : : 7 Description: number of days of the x-th month of a non-leap pyear x f(x) lgebraic form:? 3 Enumeration:

3 Truth Table Truth table. Systematic method to describe oolean function. One row for each possible input combination. N inputs 2 N rows. Proving the equivalence of two functions Prove that x 2 -=(x+)(x-) Using algebra: (you need to follow some rules) (x+)(x-) = x 2 +x-x-= x 2 - x y x y ND truth table Using enumeration: x (x+)(x-) x : : : 9 x + = x + = x x + x = x. = x x. = x.x = DeMorgan Law x.y = x + y Important laws x + y = y + x x + (y+z) = (x+y) + z x.y = y.x x.(y.z) = (x.y).z x.(y+z) = xy + xz Simplifying oolean Equations Example : = + Chapter 2 <2>

4 Simplifying oolean Equations Example : = + = ( + ) = () = Simplifying oolean Equations Example 2: = ( + C) Chapter 2 <3> Chapter 2 <4> Simplifying oolean Equations DeMorgan s Theorem Example 2: = ( + C) = (( + C)) = (()) = () = () = = = + = + = Chapter 2 <5> Chapter 2 <6>

5 ubble Pushing ubble Pushing ackward: ody changes dds bubbles to inputs What is the oolean expression for this circuit? Forward: ody changes dds bubble to output C D Chapter 2 <7> Chapter 2 <8> ubble Pushing What is the oolean expression for this circuit? C D = + CD ubble Pushing Rules egin at output, then work toward inputs Push bubbles on final output back Draw gates in a form so bubbles cancel C D Chapter 2 <9> Chapter 2 <2>

6 ubble Pushing Example ubble Pushing Example no output bubble C D C D Chapter 2 <2> Chapter 2 <22> ubble Pushing Example ubble Pushing Example no output bubble no output bubble C D C D bubble on input and output bubble on input and output C D C D no bubble on input and output C D = C + D Chapter 2 <23> Chapter 2 <24>

7 Truth Tables ( of 3) oolean function has one or more oolean inputs, and returns a single oolean output. truth table shows all the inputs and outputs of a oolean function Example: X Truth Tables (2 of 3) Example: X Truth Tables (3 of 3) When s=, return x; otherwise, return y. Example: ( S) (X S) X S mux Z Two-input multiplexer Truth Table for Functions of 2 Variables Truth table. 6 oolean functions of 2 variables. every 4-bit value represents one x y ZERO ND x y XOR OR Truth table for all oolean functions of 2 variables x y NOR EQ y' x' NND Truth table for all oolean functions of 2 variables ONE 27 28

8 ll oolean functions of 2 variables Truth Table for Functions of 3 Variables Truth table. 6 oolean functions of 2 variables. 256 oolean functions of 3 variables. 2^(2^n) oolean functions of n variables! every 4-bit value represents one every 8-bit value represents one every 2 n -bit value represents one x y z ND OR MJ ODD some functions of 3 variables Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 29 3 Sum-of-Products Sum-of-products. Systematic procedure for representing a oolean function using ND, OR, NOT. Form ND term for each in oolean function. OR terms together. x y z MJ x'yz xy'z xyz' xyz expressing MJ using sum-of-products proves that { ND, OR, NOT } are universal x'yz + xy'z + xyz' + xyz Universality of ND, OR, NOT Fact. ny oolean function can be expressed using ND, OR, NOT. { ND, OR, NOT } are universal. Notation Ex: XOR(x,y) = xy' + x'y. x' NOT x x y x + y Expressing XOR Using ND, OR, NOT x y x' y' x'y xy' x'y + xy' x XOR y Exercise. Show {ND, NOT}, {OR, NOT}, {NND}, {NOR} are universal. Hint. DeMorgan's law: (x'y')' = x + y. Meaning x ND y x OR y 3 32

9 From Math to Real-World implementation We can implement any oolean function using NND gates only. Implementation of gates Fluid switch ( We talk about abstract oolean algebra (logic) so far. Is it possible to realize it in real world? The technology needs to permit switching and conducting. It can be built using magnetic, optical, biological, i l hydraulic and pneumatic mechanism. 33 Digital Circuits What is a digital system? nalog: signals vary continuously. Digital: signals are or. 3.3V 2.8V.5V Why digital systems?.v ccuracy and reliability. Staggeringly fast and cheap. asic abstractions. On, off. Wire: propagates on/off value. Switch: controls propagation p of on/off values through wires. Wires Wires. On (): connected to power. Off (): not connected to power. If a wire is connected to a wire that is on, that wire is also on. Typical drawing convention: "flow" from top, left to bottom, right. power connection 35 36

10 Controlled Switch Relay Controlled switch. [relay implementation] 3 connections: input, output, control. Magnetic force pulls on a contact that cuts electrical flow. Control wire affects output wire, but output does not affect control; establishes forward flow of information on over time. X X Circuit natomy Logic Gates: Fundamental uilding locks 39 4

11 NOT NOT 4 42 OR Series relays = NOR 43 44

12 OR ND ND Logic Gates: Fundamental uilding locks 47 48

13 What about parallel relays? =NND Can we implement ND/OR using parallel relays? Now we know how to implement ND,OR and NOT. We can just use them as black boxes without knowing how they were implemented. Principle of information hiding Multiway Gates Multiway gates. OR: if any input is ; otherwise. ND: if all inputs are ; otherwise. Generalized: negate some inputs. Multiway Gates Multiway gates. OR: if any input is ; otherwise. ND: if all inputs are ; otherwise. Generalized: negate some inputs. 5 52

14 Multiway Gates Multiway gates. Can also be built from 2-way ygates (less efficient but implementation independent) Example: build 4-way OR from 2-way ORs Translate oolean Formula to oolean Circuit Sum-of-products. XOR Translate oolean Formula to oolean Circuit Sum-of-products. XOR. Translate oolean Formula to oolean Circuit Sum-of-products. XOR

15 Gate logic ODD Parity Circuit Interface a Xor b out ODD(x, y, z). if odd number of inputs are. otherwise. a b out a Implementation ti nd Not Or out b Not nd Xor(a,b) = Or(nd(a,Not(b)),nd(Not(a),b))) (b)) d(n ( ) b))) ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. x y z ODD x'y'z x'yz' xy'z' xyz x'y'z+x'yz' y + xy'z' + xyz Expressing ODD using sum-of-products 59 6

16 ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. Expressing a oolean Function Using ND, OR, NOT Ingredients. ND gates. OR gates. NOT gates. Wire. Instructions. Step : represent input and output signals with oolean variables. Step 2: construct truth table to carry out computation. Step 3: derive (simplified) oolean expression using sum-of products. Step 4: transform oolean expression into circuit Translate oolean Formula to oolean Circuit Sum-of-products. Majority. Translate oolean Formula to oolean Circuit Sum-of-products. Majority

17 Translate oolean Formula to oolean Circuit Sum-of-products. Majority. Translate oolean Formula to oolean Circuit Sum-of-products. Majority Simplification Using oolean lgebra Every function can be written as sum-of-product oolean expression simplification Karnaugh map Many possible circuits for each oolean function. Sum-of-products not necessarily optimal in: number of switches (space) depth of circuit (time) 67 68

18 Karnaugh Maps (K Maps) oolean expressions can be minimized by combining terms K maps minimize equations graphically P + P = P C C C C C C C C C C C K Map Circle s in adjacent squares In oolean expression, include only literals whose true and complement form are not in the circle C C = Chapter 2 <69> Chapter 2 <7> 3 Input K Map 3 Input K Map C C C C C C C C C C C C C C C C C C Truth Table C K-Map C Chapter 2 <7> Truth Table K-Map C C = + C Chapter 2 <72>

19 K Map Rules Every must be circled at least once Each circle must span a power of 2 (i.e., 2, 4) squares in each direction Each circle must be as large as possible circle may wrap around the edges don't care (X) is circled only if it helps minimize the equation 4 Input K Map C D CD Chapter 2 <73> Chapter 2 <74> 4 Input K Map 4 Input K Map C D CD C D CD = C + D + C + D Chapter 2 <75> Chapter 2 <76>

20 K Maps with Don t Cares K Maps with Don t Cares C D CD X X X X X X X C D X X X X X X X CD X X X X X X X Chapter 2 <77> Chapter 2 <78> K Maps with Don t Cares Example C D X X X X X X X CD X X X X X X X = + D + C xy z Chapter 2 <79> 8

21 Simplification Using oolean lgebra Many possible circuits for each oolean function. Sum-of-products not necessarily optimal in: number of switches (space) depth of circuit (time) MJ(x, y, z) = x'yz + xy'z + xyz' + xyz = xy + yz + xz. Layers of abstraction. Layers of bstraction uild a circuit from wires and switches. [implementation] Define a circuit by its inputs and outputs. [PI] To control complexity, encapsulate circuits. [DT] 8 82 Layers of bstraction Layers of abstraction. uild a circuit from wires and switches. [implementation] Define a circuit by its inputs and outputs. [PI] To control complexity, encapsulate circuits. [DT] Specification Step : identify input and output t Step 2: construct truth table Step 3: derive (simplified) oolean expression using sum-of products. Step 4: transform oolean expression into circuit/implement it using HDL. ou would like to test the gate before packaging. 83

22 HDL Example: uilding an nd gate a b nd out nd.cmp a b out Contract: When running your.hdl on our.tst, your.out should be the same as our.cmp. nd.hdl nd.tst CHIP nd load nd.hdl, { IN a, b; output-file nd.out, OUT out; compare-to nd.cmp, // implementation missing output-list a b out; } set a,set b,eval,output;, set a,set b,eval,output; set a,set b,eval,output; set a, set b, eval, output; Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 86 uilding an nd gate uilding an nd gate Interface: nd(a,b) = exactly when a=b= Implementation: nd(a,b) = Not(Nand(a,b)) a b nd out a b out nd.hdl CHIP nd { IN a, b; OUT out; // implementation missing } nd.hdl CHIP nd { IN a, b; OUT out; // implementation missing } Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 87 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 88

23 uilding an nd gate uilding an nd gate Implementation: nd(a,b) = Not(Nand(a,b)) Implementation: nd(a,b) = Not(Nand(a,b)) a b a b Nand out x in Not out out a a out NND x in NOT out b b out nd.hdl nd.hdl CHIP nd { IN a, b; OUT out; // implementation missing } CHIP nd { IN a, b; OUT out; Nand(a = a, b = b, out = x); Not(in = x, out = out) } Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 89 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 9 Hardware simulator (demonstrating Xor gate construction) Hardware simulator HDL program test script HDL program Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 9 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 92

24 Hardware simulator Project materials: Project web site HDL program output file nd.hdl, nd.tst, nd.cmp files Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 93 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 94 Project tips Gates for project # (asic Gates) Read the Introduction + Chapter of the book Download the book s software suite Go through the hardware simulator tutorial Do Project (optional) ou re in business. Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 95 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 96

25 Gates for project # Gates for project # (Multi-bit version) Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 97 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 98 Gates for project # (Multi-way version) Gates for project # (Multi-way version) Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 99 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide

26 Gates for project # (Multi-way version) Gates for project # (Multi-way version) Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 2 Perspective Each oolean function has a canonical representation The canonical representation is expressed in terms of nd, Not, Or End notes: Canonical representation Whodunit story: Each suspect may or may not have an alibi (a), a motivation to commit the crime (m), and a relationship to the weapon found in the scene of the crime (w). The police decides to focus attention only on suspects for whom the proposition Not(a) nd (m Or w) is true. nd, Not, Or can be expressed in terms of Nand alone Ergo, every oolean function can be realized by a standard PLD consisting of Nand gates only Truth table of the "suspect" function s( a, m, w) a ( m w) Mass production a b c and Universal building blocks, unique topology. or f(a,b,c) Gates, neurons, atoms, and Canonical form: s( a, m, w) a m w a mw a mw Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 3 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 4

27 End notes: Canonical representation (cont.) End notes: Programmable Logic Device for 3-way functions a m w s( a, m, w) a ( m w) or and s s( a, m, w) a m w a mw a mw a b c 8 and terms connected to the same 3 inputs and. legend: active fuse blown fuse or f(a,b,c) a m w and and or s and single or term connected to the outputs of 8 and terms and _ PLD implementation of f(a,b,c)= a b c + a b c (the on/off states of the fuses determine which gates participate in the computation) Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 5 Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 6 End notes: Programmable Logic Device for 3-way functions Two-level logic: NDs followed by ORs Example: = C + C + C C C minterm: C minterm: C minterm: C Elements of Computing Systems, Nisan & Schocken, MIT Press, Chapter : oolean Logic slide 7

6. Combinational Circuits. Building Blocks. Digital Circuits. Wires. Q. What is a digital system? A. Digital: signals are 0 or 1.

6. Combinational Circuits. Building Blocks. Digital Circuits. Wires. Q. What is a digital system? A. Digital: signals are 0 or 1. Digital Circuits 6 Combinational Circuits Q What is a digital system? A Digital: signals are or analog: signals vary continuously Q Why digital systems? A Accurate, reliable, fast, cheap Basic abstractions

More information

Combinational Circuits

Combinational Circuits Combinational Circuits Q. What is a combinational circuit? A. Digital: signals are or. A. No feedback: no loops. analog circuits: signals vary continuously sequential circuits: loops allowed (stay tuned)

More information

Combinational Circuits Digital Logic (Materials taken primarily from:

Combinational Circuits Digital Logic (Materials taken primarily from: Combinational Circuits Digital Logic (Materials taken primarily from: http://www.facstaff.bucknell.edu/mastascu/elessonshtml/eeindex.html http://www.cs.princeton.edu/~cos126 ) Digital Systems What is a

More information

Chapter 3. Gate-Level Minimization. Outlines

Chapter 3. Gate-Level Minimization. Outlines Chapter 3 Gate-Level Minimization Introduction The Map Method Four-Variable Map Five-Variable Map Outlines Product of Sums Simplification Don t-care Conditions NAND and NOR Implementation Other Two-Level

More information

Combinational Logic & Circuits

Combinational Logic & Circuits Week-I Combinational Logic & Circuits Spring' 232 - Logic Design Page Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other

More information

6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( )

6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( ) 6. Combinational Circuits George Boole (85 864) Claude Shannon (96 2) Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate logical values from place to place.

More information

6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( )

6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( ) 6. Combinational Circuits George Boole (85 864) Claude Shannon (96 2) Signals and Wires Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate digital signals

More information

2/8/2017. SOP Form Gives Good Performance. ECE 120: Introduction to Computing. K-Maps Can Identify Single-Gate Functions

2/8/2017. SOP Form Gives Good Performance. ECE 120: Introduction to Computing. K-Maps Can Identify Single-Gate Functions University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing Two-Level Logic SOP Form Gives Good Performance s you know, one can use a K-map

More information

Gate Level Minimization

Gate Level Minimization Gate Level Minimization By Dr. M. Hebaishy Digital Logic Design Ch- Simplifying Boolean Equations Example : Y = AB + AB Example 2: = B (A + A) T8 = B () T5 = B T Y = A(AB + ABC) = A (AB ( + C ) ) T8 =

More information

Gate Level Minimization Map Method

Gate Level Minimization Map Method Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically

More information

Gate-Level Minimization. BME208 Logic Circuits Yalçın İŞLER

Gate-Level Minimization. BME208 Logic Circuits Yalçın İŞLER Gate-Level Minimization BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com Complexity of Digital Circuits Directly related to the complexity of the algebraic expression we use to

More information

There are only 16 possible 2-input gates Let s examine all of them. Some we already know, others are just silly.

There are only 16 possible 2-input gates Let s examine all of them. Some we already know, others are just silly. ll the Gates There are only 6 possible 2-input gates Let s examine all of them. Some we already know, others are just silly. Do we really need all of these gates? How many of these gates can be implemented

More information

CS8803: Advanced Digital Design for Embedded Hardware

CS8803: Advanced Digital Design for Embedded Hardware CS883: Advanced Digital Design for Embedded Hardware Lecture 2: Boolean Algebra, Gate Network, and Combinational Blocks Instructor: Sung Kyu Lim (limsk@ece.gatech.edu) Website: http://users.ece.gatech.edu/limsk/course/cs883

More information

Experiment 3: Logic Simplification

Experiment 3: Logic Simplification Module: Logic Design Name:... University no:.. Group no:. Lab Partner Name: Mr. Mohamed El-Saied Experiment : Logic Simplification Objective: How to implement and verify the operation of the logical functions

More information

Logic Design (Part 2) Combinational Logic Circuits (Chapter 3)

Logic Design (Part 2) Combinational Logic Circuits (Chapter 3) Digital Logic Circuits Logic Design (Part ) Combinational Logic Circuits (Chapter 3) ² We saw how we can build the simple logic gates using transistors ² Use these gates as building blocks to build more

More information

Gate-Level Minimization

Gate-Level Minimization Gate-Level Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method

More information

Arithmetic Logic Unit (ALU)

Arithmetic Logic Unit (ALU) Arithmetic Logic Unit (ALU) Introduction to Computer Yung-Yu Chuang with slides by Sedgewick & Wayne (introcs.cs.princeton.edu), Nisan & Schocken (www.nand2tetris.org) and Harris & Harris (DDCA) Let's

More information

QUESTION BANK FOR TEST

QUESTION BANK FOR TEST CSCI 2121 Computer Organization and Assembly Language PRACTICE QUESTION BANK FOR TEST 1 Note: This represents a sample set. Please study all the topics from the lecture notes. Question 1. Multiple Choice

More information

Chapter 2. Boolean Expressions:

Chapter 2. Boolean Expressions: Chapter 2 Boolean Expressions: A Boolean expression or a function is an expression which consists of binary variables joined by the Boolean connectives AND and OR along with NOT operation. Any Boolean

More information

Gate-Level Minimization

Gate-Level Minimization Gate-Level Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2011 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method

More information

Boolean Algebra and Logic Gates

Boolean Algebra and Logic Gates Boolean Algebra and Logic Gates Binary logic is used in all of today's digital computers and devices Cost of the circuits is an important factor Finding simpler and cheaper but equivalent circuits can

More information

University of Technology

University of Technology University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year Lecture 5 & 6 Minimization with Karnaugh Maps Karnaugh maps lternate way of representing oolean function ll rows

More information

Philadelphia University Faculty of Information Technology Department of Computer Science. Computer Logic Design. By Dareen Hamoudeh.

Philadelphia University Faculty of Information Technology Department of Computer Science. Computer Logic Design. By Dareen Hamoudeh. Philadelphia University Faculty of Information Technology Department of Computer Science Computer Logic Design By Dareen Hamoudeh Dareen Hamoudeh 1 Canonical Forms (Standard Forms of Expression) Minterms

More information

Review: Standard forms of expressions

Review: Standard forms of expressions Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and NOT. These operations can be combined to form complex expressions, which can

More information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard

More information

Experiment 4 Boolean Functions Implementation

Experiment 4 Boolean Functions Implementation Experiment 4 Boolean Functions Implementation Introduction: Generally you will find that the basic logic functions AND, OR, NAND, NOR, and NOT are not sufficient to implement complex digital logic functions.

More information

Simplification of Boolean Functions

Simplification of Boolean Functions Simplification of Boolean Functions Contents: Why simplification? The Map Method Two, Three, Four and Five variable Maps. Simplification of two, three, four and five variable Boolean function by Map method.

More information

Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic

Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic Question 1: Due October 19 th, 2009 A convenient shorthand for specifying

More information

Boolean Logic CS.352.F12

Boolean Logic CS.352.F12 Boolean Logic CS.352.F12 Boolean Algebra Boolean Algebra Mathematical system used to manipulate logic equations. Boolean: deals with binary values (True/False, yes/no, on/off, 1/0) Algebra: set of operations

More information

Get Free notes at Module-I One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)

More information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian محرم 1439 ه Winter

More information

Circuit analysis summary

Circuit analysis summary Boolean Algebra Circuit analysis summary After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. You can easily convert

More information

Chapter 2: Combinational Systems

Chapter 2: Combinational Systems Uchechukwu Ofoegbu Chapter 2: Combinational Systems Temple University Adapted from Alan Marcovitz s Introduction to Logic and Computer Design Riddle Four switches can be turned on or off. One is the switch

More information

ENGIN 112 Intro to Electrical and Computer Engineering

ENGIN 112 Intro to Electrical and Computer Engineering ENGIN 2 Intro to Electrical and Computer Engineering Lecture 8 Minimization with Karnaugh Maps Overview K-maps: an alternate approach to representing oolean functions K-map representation can be used to

More information

Standard Forms of Expression. Minterms and Maxterms

Standard Forms of Expression. Minterms and Maxterms Standard Forms of Expression Minterms and Maxterms Standard forms of expressions We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains:

More information

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2008

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2008 Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 28 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom

More information

BOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits.

BOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits. COURSE / CODE DIGITAL SYSTEMS FUNDAMENTAL (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) BOOLEAN ALGEBRA Boolean Logic Boolean logic is a complete system for logical operations. It is used in countless

More information

1. Mark the correct statement(s)

1. Mark the correct statement(s) 1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another

More information

Chapter 2 Boolean algebra and Logic Gates

Chapter 2 Boolean algebra and Logic Gates Chapter 2 Boolean algebra and Logic Gates 2. Introduction In working with logic relations in digital form, we need a set of rules for symbolic manipulation which will enable us to simplify complex expressions

More information

Gate-Level Minimization

Gate-Level Minimization MEC520 디지털공학 Gate-Level Minimization Jee-Hwan Ryu School of Mechanical Engineering Gate-Level Minimization-The Map Method Truth table is unique Many different algebraic expression Boolean expressions may

More information

DKT 122/3 DIGITAL SYSTEM 1

DKT 122/3 DIGITAL SYSTEM 1 Company LOGO DKT 122/3 DIGITAL SYSTEM 1 BOOLEAN ALGEBRA (PART 2) Boolean Algebra Contents Boolean Operations & Expression Laws & Rules of Boolean algebra DeMorgan s Theorems Boolean analysis of logic circuits

More information

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University Logic Design First Stage Lecture No.6 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Outlines Boolean Operations Laws of Boolean Algebra Rules of

More information

Chapter 3. Boolean Algebra and Digital Logic

Chapter 3. Boolean Algebra and Digital Logic Chapter 3 Boolean Algebra and Digital Logic Chapter 3 Objectives Understand the relationship between Boolean logic and digital computer circuits. Learn how to design simple logic circuits. Understand how

More information

Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions

Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions 1. Convert the following SOP expression to an equivalent POS expression. 2. Determine the values of A, B, C, and D that make

More information

Boolean Algebra. BME208 Logic Circuits Yalçın İŞLER

Boolean Algebra. BME208 Logic Circuits Yalçın İŞLER Boolean Algebra BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com 5 Boolean Algebra /2 A set of elements B There exist at least two elements x, y B s. t. x y Binary operators: +

More information

Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples

Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples Lecture B: Logic Minimization Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples Incompletely specified functions

More information

Menu. Algebraic Simplification - Boolean Algebra EEL3701 EEL3701. MSOP, MPOS, Simplification

Menu. Algebraic Simplification - Boolean Algebra EEL3701 EEL3701. MSOP, MPOS, Simplification Menu Minterms & Maxterms SOP & POS MSOP & MPOS Simplification using the theorems/laws/axioms Look into my... 1 Definitions (Review) Algebraic Simplification - Boolean Algebra Minterms (written as m i ):

More information

Digital Logic Design (3)

Digital Logic Design (3) Digital Logic Design (3) ENGG1015 1 st Semester, 2010 Dr. Kenneth Wong Dr. Hayden So Department of Electrical and Electronic Engineering Last lecture ll logic functions can be represented as (1) truth

More information

Digital Logic Lecture 7 Gate Level Minimization

Digital Logic Lecture 7 Gate Level Minimization Digital Logic Lecture 7 Gate Level Minimization By Ghada Al-Mashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. K-map principles. Simplification using K-maps. Don t-care

More information

IT 201 Digital System Design Module II Notes

IT 201 Digital System Design Module II Notes IT 201 Digital System Design Module II Notes BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.

More information

6: Combinational Circuits

6: Combinational Circuits Computer Architecture 6: Combinational Circuits Previous two lectures. von Neumann machine. This lectures. Boolean circuits. Later in the course. Putting it all together and building a TOY machine. George

More information

Unit-IV Boolean Algebra

Unit-IV Boolean Algebra Unit-IV Boolean Algebra Boolean Algebra Chapter: 08 Truth table: Truth table is a table, which represents all the possible values of logical variables/statements along with all the possible results of

More information

Chapter 2 Combinational

Chapter 2 Combinational Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization HOANG Trang 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean Equations Binary Logic

More information

Objectives: 1- Bolean Algebra. Eng. Ayman Metwali

Objectives: 1- Bolean Algebra. Eng. Ayman Metwali Objectives: Chapter 3 : 1- Boolean Algebra Boolean Expressions Boolean Identities Simplification of Boolean Expressions Complements Representing Boolean Functions 2- Logic gates 3- Digital Components 4-

More information

2.1 Binary Logic and Gates

2.1 Binary Logic and Gates 1 EED2003 Digital Design Presentation 2: Boolean Algebra Asst. Prof.Dr. Ahmet ÖZKURT Asst. Prof.Dr Hakkı T. YALAZAN Based on the Lecture Notes by Jaeyoung Choi choi@comp.ssu.ac.kr Fall 2000 2.1 Binary

More information

Gate-Level Minimization. section instructor: Ufuk Çelikcan

Gate-Level Minimization. section instructor: Ufuk Çelikcan Gate-Level Minimization section instructor: Ufuk Çelikcan Compleity of Digital Circuits Directly related to the compleity of the algebraic epression we use to build the circuit. Truth table may lead to

More information

Digital Fundamentals

Digital Fundamentals Digital Fundamentals Tenth Edition Floyd Chapter 3 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, th 28 Pearson Education ENE, KMUTT ed 29 The Inverter Summary The inverter performs

More information

SWITCHING THEORY AND LOGIC CIRCUITS

SWITCHING THEORY AND LOGIC CIRCUITS SWITCHING THEORY AND LOGIC CIRCUITS COURSE OBJECTIVES. To understand the concepts and techniques associated with the number systems and codes 2. To understand the simplification methods (Boolean algebra

More information

Propositional Calculus: Boolean Algebra and Simplification. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson

Propositional Calculus: Boolean Algebra and Simplification. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus: Boolean Algebra and Simplification CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Topics Motivation: Simplifying Conditional Expressions

More information

UNIT-4 BOOLEAN LOGIC. NOT Operator Operates on single variable. It gives the complement value of variable.

UNIT-4 BOOLEAN LOGIC. NOT Operator Operates on single variable. It gives the complement value of variable. UNIT-4 BOOLEAN LOGIC Boolean algebra is an algebra that deals with Boolean values((true and FALSE). Everyday we have to make logic decisions: Should I carry the book or not?, Should I watch TV or not?

More information

Lecture (05) Boolean Algebra and Logic Gates

Lecture (05) Boolean Algebra and Logic Gates Lecture (05) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Minterms and Maxterms consider two binary variables x and y combined with an AND operation. Since eachv ariable may appear in either

More information

Simplification of Boolean Functions

Simplification of Boolean Functions COM111 Introduction to Computer Engineering (Fall 2006-2007) NOTES 5 -- page 1 of 5 Introduction Simplification of Boolean Functions You already know one method for simplifying Boolean expressions: Boolean

More information

Ch. 5 : Boolean Algebra &

Ch. 5 : Boolean Algebra & Ch. 5 : Boolean Algebra & Reduction elektronik@fisika.ui.ac.id Objectives Should able to: Write Boolean equations for combinational logic applications. Utilize Boolean algebra laws and rules for simplifying

More information

Announcements. Chapter 2 - Part 1 1

Announcements. Chapter 2 - Part 1 1 Announcements If you haven t shown the grader your proof of prerequisite, please do so by 11:59 pm on 09/05/2018 (Wednesday). I will drop students that do not show us the prerequisite proof after this

More information

Summary. Boolean Addition

Summary. Boolean Addition Summary Boolean Addition In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. A single variable can only have a value of or 0. The complement represents the inverse

More information

Computer Science. Unit-4: Introduction to Boolean Algebra

Computer Science. Unit-4: Introduction to Boolean Algebra Unit-4: Introduction to Boolean Algebra Learning Objective At the end of the chapter students will: Learn Fundamental concepts and basic laws of Boolean algebra. Learn about Boolean expression and will

More information

Review. EECS Components and Design Techniques for Digital Systems. Lec 05 Boolean Logic 9/4-04. Seq. Circuit Behavior. Outline.

Review. EECS Components and Design Techniques for Digital Systems. Lec 05 Boolean Logic 9/4-04. Seq. Circuit Behavior. Outline. Review EECS 150 - Components and Design Techniques for Digital Systems Lec 05 Boolean Logic 94-04 David Culler Electrical Engineering and Computer Sciences University of California, Berkeley Design flow

More information

X Y Z F=X+Y+Z

X Y Z F=X+Y+Z This circuit is used to obtain the compliment of a value. If X = 0, then X = 1. The truth table for NOT gate is : X X 0 1 1 0 2. OR gate : The OR gate has two or more input signals but only one output

More information

Combinational Logic Circuits

Combinational Logic Circuits Chapter 2 Combinational Logic Circuits J.J. Shann (Slightly trimmed by C.P. Chung) Chapter Overview 2-1 Binary Logic and Gates 2-2 Boolean Algebra 2-3 Standard Forms 2-4 Two-Level Circuit Optimization

More information

Module -7. Karnaugh Maps

Module -7. Karnaugh Maps 1 Module -7 Karnaugh Maps 1. Introduction 2. Canonical and Standard forms 2.1 Minterms 2.2 Maxterms 2.3 Canonical Sum of Product or Sum-of-Minterms (SOM) 2.4 Canonical product of sum or Product-of-Maxterms(POM)

More information

2.6 BOOLEAN FUNCTIONS

2.6 BOOLEAN FUNCTIONS 2.6 BOOLEAN FUNCTIONS Binary variables have two values, either 0 or 1. A Boolean function is an expression formed with binary variables, the two binary operators AND and OR, one unary operator NOT, parentheses

More information

S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017

S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017 S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017 Karnaugh Map Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and

More information

Software Engineering 2DA4. Slides 2: Introduction to Logic Circuits

Software Engineering 2DA4. Slides 2: Introduction to Logic Circuits Software Engineering 2DA4 Slides 2: Introduction to Logic Circuits Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on S. Brown and Z. Vranesic, Fundamentals of Digital

More information

Lecture 10: Combinational Circuits

Lecture 10: Combinational Circuits Computer Architecture Lecture : Combinational Circuits Previous two lectures.! TOY machine. Net two lectures.! Digital circuits. George Boole (85 864) Claude Shannon (96 2) Culminating lecture.! Putting

More information

Switching Circuits & Logic Design

Switching Circuits & Logic Design Switching Circuits & Logic Design Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 23 5 Karnaugh Maps K-map Walks and Gray Codes http://asicdigitaldesign.wordpress.com/28/9/26/k-maps-walks-and-gray-codes/

More information

CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey

CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input

More information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian ذو الحجة 1438 ه Winter

More information

CS470: Computer Architecture. AMD Quad Core

CS470: Computer Architecture. AMD Quad Core CS470: Computer Architecture Yashwant K. Malaiya, Professor malaiya@cs.colostate.edu AMD Quad Core 1 Architecture Layers Building blocks Gates, flip-flops Functional bocks: Combinational, Sequential Instruction

More information

Computer Engineering Chapter 3 Boolean Algebra

Computer Engineering Chapter 3 Boolean Algebra Computer Engineering Chapter 3 Boolean Algebra Hiroaki Kobayashi 5/30/2011 Ver. 06102011 5/30/2011 Computer Engineering 1 Agenda in Chapter 3 What is Boolean Algebra Basic Boolean/Logical Operations (Operators)

More information

Graduate Institute of Electronics Engineering, NTU. CH5 Karnaugh Maps. Lecturer: 吳安宇教授 Date:2006/10/20 ACCESS IC LAB

Graduate Institute of Electronics Engineering, NTU. CH5 Karnaugh Maps. Lecturer: 吳安宇教授 Date:2006/10/20 ACCESS IC LAB CH5 Karnaugh Maps Lecturer: 吳安宇教授 Date:2006/0/20 CCESS IC L Problems in lgebraic Simplification The procedures are difficult to apply in a systematic way. It is difficult to tell when you have arrived

More information

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2006

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2006 Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 26 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom

More information

A B AB CD Objectives:

A B AB CD Objectives: Objectives:. Four variables maps. 2. Simplification using prime implicants. 3. "on t care" conditions. 4. Summary.. Four variables Karnaugh maps Minterms A A m m m3 m2 A B C m4 C A B C m2 m8 C C m5 C m3

More information

Code No: 07A3EC03 Set No. 1

Code No: 07A3EC03 Set No. 1 Code No: 07A3EC03 Set No. 1 II B.Tech I Semester Regular Examinations, November 2008 SWITCHING THEORY AND LOGIC DESIGN ( Common to Electrical & Electronic Engineering, Electronics & Instrumentation Engineering,

More information

Points Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 4: Logic Simplication & Karnaugh Map

Points Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 4: Logic Simplication & Karnaugh Map Points Addressed in this Lecture Lecture 4: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London Standard form of Boolean Expressions Sum-of-Products (SOP),

More information

UNIT 2 BOOLEAN ALGEBRA

UNIT 2 BOOLEAN ALGEBRA UNIT 2 BOOLEN LGEBR Spring 2 2 Contents Introduction Basic operations Boolean expressions and truth tables Theorems and laws Basic theorems Commutative, associative, and distributive laws Simplification

More information

Specifying logic functions

Specifying logic functions CSE4: Components and Design Techniques for Digital Systems Specifying logic functions Instructor: Mohsen Imani Slides from: Prof.Tajana Simunic and Dr.Pietro Mercati We have seen various concepts: Last

More information

Chapter 2. Boolean Algebra and Logic Gates

Chapter 2. Boolean Algebra and Logic Gates Chapter 2. Boolean Algebra and Logic Gates Tong In Oh 1 Basic Definitions 2 3 2.3 Axiomatic Definition of Boolean Algebra Boolean algebra: Algebraic structure defined by a set of elements, B, together

More information

Propositional Calculus. Math Foundations of Computer Science

Propositional Calculus. Math Foundations of Computer Science Propositional Calculus Math Foundations of Computer Science Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus so that they can use it to

More information

Boolean algebra. June 17, Howard Huang 1

Boolean algebra. June 17, Howard Huang 1 Boolean algebra Yesterday we talked about how analog voltages can represent the logical values true and false. We introduced the basic Boolean operations AND, OR and NOT, which can be implemented in hardware

More information

2008 The McGraw-Hill Companies, Inc. All rights reserved.

2008 The McGraw-Hill Companies, Inc. All rights reserved. 28 The McGraw-Hill Companies, Inc. All rights reserved. 28 The McGraw-Hill Companies, Inc. All rights reserved. All or Nothing Gate Boolean Expression: A B = Y Truth Table (ee next slide) or AB = Y 28

More information

Variable, Complement, and Literal are terms used in Boolean Algebra.

Variable, Complement, and Literal are terms used in Boolean Algebra. We have met gate logic and combination of gates. Another way of representing gate logic is through Boolean algebra, a way of algebraically representing logic gates. You should have already covered the

More information

數位系統 Digital Systems 朝陽科技大學資工系. Speaker: Fuw-Yi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷

數位系統 Digital Systems 朝陽科技大學資工系. Speaker: Fuw-Yi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷 數位系統 Digital Systems Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝陽科技大學資工系 Speaker: Fuw-Yi Yang 楊伏夷 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象,

More information

Combinational Logic Circuits Part III -Theoretical Foundations

Combinational Logic Circuits Part III -Theoretical Foundations Combinational Logic Circuits Part III -Theoretical Foundations Overview Simplifying Boolean Functions Algebraic Manipulation Karnaugh Map Manipulation (simplifying functions of 2, 3, 4 variables) Systematic

More information

Introduction to Boolean logic and Logical Gates

Introduction to Boolean logic and Logical Gates Introduction to Boolean logic and Logical Gates Institute of Statistics Fall 2014 We saw the importance of the binary number system for data representation in a computer system. We ll see that the construction

More information

Digital Logic Design. Outline

Digital Logic Design. Outline Digital Logic Design Gate-Level Minimization CSE32 Fall 2 Outline The Map Method 2,3,4 variable maps 5 and 6 variable maps (very briefly) Product of sums simplification Don t Care conditions NAND and NOR

More information

7/25/2016. Example: Addition of Unsigned Bit Patterns. ECE 120: Introduction to Computing. Adding Two Non-Negative Patterns Can Overflow

7/25/2016. Example: Addition of Unsigned Bit Patterns. ECE 120: Introduction to Computing. Adding Two Non-Negative Patterns Can Overflow University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing 2 s Complement Overflow and Boolean Logic Example: ddition of Unsigned Bit Patterns

More information

Lecture 7 Logic Simplification

Lecture 7 Logic Simplification Lecture 7 Logic Simplification Simplification Using oolean lgebra simplified oolean expression uses the fewest gates possible to implement a given expression. +(+)+(+) Simplification Using oolean lgebra

More information

Lecture 4: Implementation AND, OR, NOT Gates and Complement

Lecture 4: Implementation AND, OR, NOT Gates and Complement EE210: Switching Systems Lecture 4: Implementation AND, OR, NOT Gates and Complement Prof. YingLi Tian Feb. 13, 2018 Department of Electrical Engineering The City College of New York The City University

More information

Combinational Logic Circuits

Combinational Logic Circuits Chapter 3 Combinational Logic Circuits 12 Hours 24 Marks 3.1 Standard representation for logical functions Boolean expressions / logic expressions / logical functions are expressed in terms of logical

More information

ece5745-pla-notes.txt

ece5745-pla-notes.txt ece5745-pla-notes.txt ========================================================================== Follow up on PAL/PROM/PLA Activity ==========================================================================

More information