Module 1: Propositional Logic. CPSC 121: Models of Computation. Module 1: Coming up... Module 1: Coming up...
|
|
|
- Louisa Dickerson
- 7 years ago
- Views:
Transcription
1 CPSC 121: Models of Computation By the start of the class, you should be able to: Translate back and forth between simple natural language statements and propositional logic. Evaluate the truth of propositional logic statements using truth tables. Translate back and forth between propositional logic statements and circuits that assess the truth of those statements. CPSC W T1 2 Module 1: Coming up... Module 1: Coming up... What is coming up Second pre-class quiz: due tomorrow at 19:00. Assigned reading for the quiz: Epp, 4 th edition: 2.2 Epp, 3 rd edition: 1.2 Rosen, 6 th or 7 th edition: 1.1 from page 6 onwards. Assignment #1 due Monday September 28 th, 2015 at 17:00. assignment box #21 is in room ICCS X235. CPSC W T1 3 What is coming up Third pre-class quiz: due Wednesday September 23 rd at 19:00. Assigned reading for the quiz: Epp, 4th edition: 2.5 Epp, 3rd edition: 1.5 Rosen, any edition: not much Also read: CPSC W T1 4
2 CPSC W T1 6 CPSC 121: the BIG questions: We are not yet ready to directly address any of the big questions. But this module lays out the ground work for all of them. By the end of this module, you should be able to: Build computational systems to solve real problems, using both propositional logic expressions and equivalent digital logic circuits, The light switches problem from the 1st pre-class quiz. The 7- or 4-segment LED displays we will discuss in class. CPSC W T1 7 Module 1.1: From circuits to propositions Module outline: From circuits to propositions Light switches 7-segment displays More exercises How do we find the logical expression that corresponds to a circuit's output First we write the operator for the gate that produces the circuit's output. The circuit for the first of that gate's inputs will be the operator's left argument. The circuit for the second of that gate's inputs will be the operator's right argument. This is our first algorithm! CPSC W T1 8 CPSC W T1 9
3 Module 1.1: From circuits to propositions Module 1.1: From circuits to propositions Example: ( ( a c) v ( a b ) ) ^ ~( ( a b) ( c d) ) What does this circuit compute Note: when you write a truth table, always list the combinations in the same order. For instance, with 3 variables the first column contains 4 false followed by 4 true. the second column contains 2 false, 2 true, 2 false, 2 true. and the third column alternates false with true. With k variables, the first column has 2 k-1 false and then 2 k-1 true, the second column has 2 k-2 false and then 2 k-2 true (twice), etc. CPSC W T1 10 CPSC W T1 11 Module 1.1: From circuits to propositions What logical expression corresponds to the following circuit Module outline: From circuits to propositions Light switches 7-segment displays More exercises CPSC W T1 12 CPSC W T1 13
4 Module 1.2: Light switches Module 1.2: Light switches Consider again: Design a light that changes state whenever any of the switches that control it is flipped. Ideally your solution would work with any number! How do we approach this Start simple: 1 switch. Then try 2 switches. Then try 3 switches. Then see if we can generalize to n switches. Design a light that changes state whenever any of the switches that control it is flipped. Ideally your solution would work with any number! One switch: first decide on the inputs and output. Which of these should appear in our circuit, as either input or output a) f: the switch is flipped b) s: the switch is on c) l: the light is on d) c: the light changed state CPSC W T1 14 CPSC W T1 15 Module 1.2: Light switches Module 1.2: Light switches Two switches: make sure we understand the problem first. Is the light on or off when both switches are on a) Always on. b) Always off. c) Depends, but a correct solution should always do the same thing. d) Depends, and a correct solution might do different things at different times. Two switches: which circuit(s) work(s) CPSC W T1 16 CPSC W T1 17
5 Module 1.2: Light switches Module 1.2: Light switches Circuit design tip: if you are not sure where to start while designing a circuit, First build the truth table. Then turn it into a circuit. For the three switches problem: We can decide arbitrarily what the output is when all three switches are OFF. This determines the output for all other cases! Let's see how... CPSC W T1 18 What pattern do we observe The light is ON if Now to generalize to n switches... What do you think the answer is How can we convince ourselves that it is correct Mathematical induction CPSC W T1 19 Module outline: From circuits to propositions Light switches 7-segment displays More exercises Problem: design a circuit that displays the numbers 0 through 9 using seven LEDs (lights) in the shape illustrated below. CPSC W T1 20 CPSC W T1 21
6 How do we represent the inputs Use logical (true/false) values. Each integer represented by 1 specific combination of input values. Could we do this randomly Yes! But we won't How many of you know about binary representation Understanding: how many inputs (wires going into the circuit) should there be a) 0 b) 1 c) 4 d) 7 e) 10 f) None of the above CPSC W T1 22 CPSC W T1 23 Here's how we will do it: Number Value 1 Value 2 Value 3 Value 4 0 F F F F 1 F F F T 2 F F T F 3 F F T T 4 F T F F 5 F T F T 6 F T T F 7 F T T T 8 T F F F 9 T F F T Understanding: how many outputs (lights) are there a) 0 b) 1 c) 4 d) 7 e) 10 f) None of the above CPSC W T1 24 CPSC W T1 25
7 Let's try it out by hand... Which other person's algorithm do you need to know about a) No one else's b) Your neighbours c) The person opposite to you d) Everybody else's e) None of the above. CPSC W T1 26 Let us look at the bottom-left segment. Which is the correct truth table Number Value 1 Value 2 Value 3 Value 4 0 F F F F 1 F F F T 2 F F T F 3 F F T T 4 F T F F 5 F T F T 6 F T T F 7 F T T T 8 T F F F (a) (b) (c) (d) (e) T T T 0 0 F F F 3 2 T T F 4 6 F F T 8 8 T F T F F F T T F F F F T T T 9 T F F T F F F CPSC W T1 27 From the truth table, we can Write a proposition for each row Or the propositions together. This works for every truth table, but may give a circuit that is much larger than necessary. Let us look at the middle segment now: How many rows of the truth table contain T How can we avoid having that many terms with 4 variables each CPSC W T1 28 Looking back at the bottom-left segment: There may be a simpler proposition, which will translate into a smaller circuit. Can you find a pattern in the rows for which the segment should be on CPSC W T1 29
8 Module 1.4: More exercises Module outline: From circuits to propositions Light switches 7-segment displays More exercises CPSC W T1 30 Prove that our two solutions for the lower-left segment are not logically equivalent. You should do this by providing values for the variables, so the two propositions have different truth values. Why are they both correct solutions, despite that Finish the problem by building circuits for the other 5 segments. Design a circuit that takes three bits as input, and outputs the binary representation for their sum. CPSC W T1 31 Module 1.4: More exercises Build a circuit that displays the numbers 1 through 9 represented by four boolean values p, q, r, and s on a 4-segment boolean display. CPSC W T1 32
Logic in Computer Science: Logic Gates
Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers
Gates, Circuits, and Boolean Algebra
Gates, Circuits, and Boolean Algebra Computers and Electricity A gate is a device that performs a basic operation on electrical signals Gates are combined into circuits to perform more complicated tasks
Binary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1.
File: chap04, Chapter 04 1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1. 2. True or False? A gate is a device that accepts a single input signal and produces one
Section IV.1: Recursive Algorithms and Recursion Trees
Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller
Session 6 Number Theory
Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple
Session 7 Fractions and Decimals
Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,
Handout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
Circuits and Boolean Expressions
Circuits and Boolean Expressions Provided by TryEngineering - Lesson Focus Boolean logic is essential to understanding computer architecture. It is also useful in program construction and Artificial Intelligence.
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
Understanding Logic Design
Understanding Logic Design ppendix of your Textbook does not have the needed background information. This document supplements it. When you write add DD R0, R1, R2, you imagine something like this: R1
CS311 Lecture: Sequential Circuits
CS311 Lecture: Sequential Circuits Last revised 8/15/2007 Objectives: 1. To introduce asynchronous and synchronous flip-flops (latches and pulsetriggered, plus asynchronous preset/clear) 2. To introduce
CSE140: Midterm 1 Solution and Rubric
CSE140: Midterm 1 Solution and Rubric April 23, 2014 1 Short Answers 1.1 True or (6pts) 1. A maxterm must include all input variables (1pt) True 2. A canonical product of sums is a product of minterms
Lab 1: Full Adder 0.0
Lab 1: Full Adder 0.0 Introduction In this lab you will design a simple digital circuit called a full adder. You will then use logic gates to draw a schematic for the circuit. Finally, you will verify
CHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.
CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:
def: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.
Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true
Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
WRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
FORDHAM UNIVERSITY CISC 3593. Dept. of Computer and Info. Science Spring, 2011. Lab 2. The Full-Adder
FORDHAM UNIVERSITY CISC 3593 Fordham College Lincoln Center Computer Organization Dept. of Computer and Info. Science Spring, 2011 Lab 2 The Full-Adder 1 Introduction In this lab, the student will construct
Formal Languages and Automata Theory - Regular Expressions and Finite Automata -
Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March
COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
Mathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
Properties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
Name: Class: Date: 9. The compiler ignores all comments they are there strictly for the convenience of anyone reading the program.
Name: Class: Date: Exam #1 - Prep True/False Indicate whether the statement is true or false. 1. Programming is the process of writing a computer program in a language that the computer can respond to
Offline 1-Minesweeper is NP-complete
Offline 1-Minesweeper is NP-complete James D. Fix Brandon McPhail May 24 Abstract We use Minesweeper to illustrate NP-completeness proofs, arguments that establish the hardness of solving certain problems.
2011, The McGraw-Hill Companies, Inc. Chapter 5
Chapter 5 5.1 Processor Memory Organization The memory structure for a PLC processor consists of several areas, some of these having specific roles. With rack-based memory structures addresses are derived
Figure 8-1 Four Possible Results of Adding Two Bits
CHPTER EIGHT Combinational Logic pplications Thus far, our discussion has focused on the theoretical design issues of computer systems. We have not yet addressed any of the actual hardware you might find
6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Introduction. The Quine-McCluskey Method Handout 5 January 21, 2016. CSEE E6861y Prof. Steven Nowick
CSEE E6861y Prof. Steven Nowick The Quine-McCluskey Method Handout 5 January 21, 2016 Introduction The Quine-McCluskey method is an exact algorithm which finds a minimum-cost sum-of-products implementation
Karnaugh Maps. Circuit-wise, this leads to a minimal two-level implementation
Karnaugh Maps 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 also be directly translated
Excel & Visual Basic for Applications (VBA)
Excel & Visual Basic for Applications (VBA) The VBA Programming Environment Recording Macros Working with the Visual Basic Editor (VBE) 1 Why get involved with this programming business? If you can't program,
EE 261 Introduction to Logic Circuits. Module #2 Number Systems
EE 261 Introduction to Logic Circuits Module #2 Number Systems Topics A. Number System Formation B. Base Conversions C. Binary Arithmetic D. Signed Numbers E. Signed Arithmetic F. Binary Codes Textbook
BOOLEAN ALGEBRA & LOGIC GATES
BOOLEAN ALGEBRA & LOGIC GATES Logic gates are electronic circuits that can be used to implement the most elementary logic expressions, also known as Boolean expressions. The logic gate is the most basic
United States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1
United States Naval Academy Electrical and Computer Engineering Department EC262 Exam 29 September 2. Do a page check now. You should have pages (cover & questions). 2. Read all problems in their entirety.
Lecture Notes on Linear Search
Lecture Notes on Linear Search 15-122: Principles of Imperative Computation Frank Pfenning Lecture 5 January 29, 2013 1 Introduction One of the fundamental and recurring problems in computer science is
Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system
CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical
Lecture Notes in Discrete Mathematics. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Lecture Notes in Discrete Mathematics Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 Preface This book is designed for a one semester course in discrete mathematics for sophomore or junior
Don t Slow Me Down with that Calculator Cliff Petrak (Teacher Emeritus) Brother Rice H.S. Chicago cpetrak1@hotmail.com
Don t Slow Me Down with that Calculator Cliff Petrak (Teacher Emeritus) Brother Rice H.S. Chicago cpetrak1@hotmail.com In any computation, we have four ideal objectives to meet: 1) Arriving at the correct
Programming A PLC. Standard Instructions
Programming A PLC STEP 7-Micro/WIN32 is the program software used with the S7-2 PLC to create the PLC operating program. STEP 7 consists of a number of instructions that must be arranged in a logical order
MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
Bits Superposition Quantum Parallelism
7-Qubit Quantum Computer Typical Ion Oscillations in a Trap Bits Qubits vs Each qubit can represent both a or at the same time! This phenomenon is known as Superposition. It leads to Quantum Parallelism
E XPLORING QUADRILATERALS
E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this
Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
Regions in a circle. 7 points 57 regions
Regions in a circle 1 point 1 region points regions 3 points 4 regions 4 points 8 regions 5 points 16 regions The question is, what is the next picture? How many regions will 6 points give? There's an
Lecture 8: Synchronous Digital Systems
Lecture 8: Synchronous Digital Systems The distinguishing feature of a synchronous digital system is that the circuit only changes in response to a system clock. For example, consider the edge triggered
Specimen 2015 am/pm Time allowed: 1hr 30mins
SPECIMEN MATERIAL GCSE COMPUTER SCIENCE 8520/1 Paper 1 Specimen 2015 am/pm Time allowed: 1hr 30mins Materials There are no additional materials required for this paper. Instructions Use black ink or black
COMPUTER SCIENCE TRIPOS
CST.98.5.1 COMPUTER SCIENCE TRIPOS Part IB Wednesday 3 June 1998 1.30 to 4.30 Paper 5 Answer five questions. No more than two questions from any one section are to be answered. Submit the answers in five
2 SYSTEM DESCRIPTION TECHNIQUES
2 SYSTEM DESCRIPTION TECHNIQUES 2.1 INTRODUCTION Graphical representation of any process is always better and more meaningful than its representation in words. Moreover, it is very difficult to arrange
Boolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
What is a Loop? Pretest Loops in C++ Types of Loop Testing. Count-controlled loops. Loops can be...
What is a Loop? CSC Intermediate Programming Looping A loop is a repetition control structure It causes a single statement or a group of statements to be executed repeatedly It uses a condition to control
Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
Chapter 1: Digital Systems and Binary Numbers
Chapter 1: Digital Systems and Binary Numbers Digital age and information age Digital computers general purposes many scientific, industrial and commercial applications Digital systems telephone switching
Logic gates. Chapter. 9.1 Logic gates. MIL symbols. Learning Summary. In this chapter you will learn about: Logic gates
Chapter 9 Logic gates Learning Summary In this chapter you will learn about: Logic gates Truth tables Logic circuits/networks In this chapter we will look at how logic gates are used and how truth tables
Boolean Algebra. Boolean Algebra. Boolean Algebra. Boolean Algebra
2 Ver..4 George Boole was an English mathematician of XIX century can operate on logic (or Boolean) variables that can assume just 2 values: /, true/false, on/off, closed/open Usually value is associated
Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic
Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1
Sources: On the Web: Slides will be available on:
C programming Introduction The basics of algorithms Structure of a C code, compilation step Constant, variable type, variable scope Expression and operators: assignment, arithmetic operators, comparison,
The Peruvian coin flip Cryptographic protocols
Activity 17 The Peruvian coin flip Cryptographic protocols Age group Older elementary and up. Abilities assumed Requires counting, and recognition of odd and even numbers. Some understanding of the concepts
(Refer Slide Time: 00:01:16 min)
Digital Computer Organization Prof. P. K. Biswas Department of Electronic & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture No. # 04 CPU Design: Tirning & Control
Take-Home Exercise. z y x. Erik Jonsson School of Engineering and Computer Science. The University of Texas at Dallas
Take-Home Exercise Assume you want the counter below to count mod-6 backward. That is, it would count 0-5-4-3-2-1-0, etc. Assume it is reset on startup, and design the wiring to make the counter count
COMPUTER SCIENCE. Paper 1 (THEORY)
COMPUTER SCIENCE Paper 1 (THEORY) (Three hours) Maximum Marks: 70 (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time) -----------------------------------------------------------------------------------------------------------------------
CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
A Prime Investigation with 7, 11, and 13
. Objective To investigate the divisibility of 7, 11, and 13, and discover the divisibility characteristics of certain six-digit numbers A c t i v i t y 3 Materials TI-73 calculator A Prime Investigation
Programmable Logic Controllers Definition. Programmable Logic Controllers History
Definition A digitally operated electronic apparatus which uses a programmable memory for the internal storage of instructions for implementing specific functions such as logic, sequencing, timing, counting,
Digital circuits make up all computers and computer systems. The operation of digital circuits is based on
Digital Logic Circuits Digital circuits make up all computers and computer systems. The operation of digital circuits is based on Boolean algebra, the mathematics of binary numbers. Boolean algebra is
Today s topics. Digital Computers. More on binary. Binary Digits (Bits)
Today s topics! Binary Numbers! Brookshear.-.! Slides from Prof. Marti Hearst of UC Berkeley SIMS! Upcoming! Networks Interactive Introduction to Graph Theory http://www.utm.edu/cgi-bin/caldwell/tutor/departments/math/graph/intro
Cosmological Arguments for the Existence of God S. Clarke
Cosmological Arguments for the Existence of God S. Clarke [Modified Fall 2009] 1. Large class of arguments. Sometimes they get very complex, as in Clarke s argument, but the basic idea is simple. Lets
Conversion Between Analog and Digital Signals
ELET 3156 DL - Laboratory #6 Conversion Between Analog and Digital Signals There is no pre-lab work required for this experiment. However, be sure to read through the assignment completely prior to starting
Two's Complement Adder/Subtractor Lab L03
Two's Complement Adder/Subtractor Lab L03 Introduction Computers are usually designed to perform indirect subtraction instead of direct subtraction. Adding -B to A is equivalent to subtracting B from A,
DEDUCTIVE & INDUCTIVE REASONING
DEDUCTIVE & INDUCTIVE REASONING Expectations 1. Take notes on inductive and deductive reasoning. 2. This is an information based presentation -- I simply want you to be able to apply this information to
Lecture 7: NP-Complete Problems
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit
CALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents
Mathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
Chapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
Karnaugh Maps & Combinational Logic Design. ECE 152A Winter 2012
Karnaugh Maps & Combinational Logic Design ECE 52A Winter 22 Reading Assignment Brown and Vranesic 4 Optimized Implementation of Logic Functions 4. Karnaugh Map 4.2 Strategy for Minimization 4.2. Terminology
Introduction to formal semantics -
Introduction to formal semantics - Introduction to formal semantics 1 / 25 structure Motivation - Philosophy paradox antinomy division in object und Meta language Semiotics syntax semantics Pragmatics
2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
Gates. J. Robert Jump Department of Electrical And Computer Engineering Rice University Houston, TX 77251
Gates J. Robert Jump Department of Electrical And Computer Engineering Rice University Houston, T 77251 1. The Evolution of Electronic Digital Devices...1 2. Logical Operations and the Behavior of Gates...2
Basic Logic Gates Richard E. Haskell
BASIC LOGIC GATES 1 E Basic Logic Gates Richard E. Haskell All digital systems are made from a few basic digital circuits that we call logic gates. These circuits perform the basic logic functions that
Counters and Decoders
Physics 3330 Experiment #10 Fall 1999 Purpose Counters and Decoders In this experiment, you will design and construct a 4-bit ripple-through decade counter with a decimal read-out display. Such a counter
earlier in the semester: The Full adder above adds two bits and the output is at the end. So if we do this eight times, we would have an 8-bit adder.
The circuit created is an 8-bit adder. The 8-bit adder adds two 8-bit binary inputs and the result is produced in the output. In order to create a Full 8-bit adder, I could use eight Full -bit adders and
CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS
CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS CENTRAL LIMIT THEOREM (SECTION 7.2 OF UNDERSTANDABLE STATISTICS) The Central Limit Theorem says that if x is a random variable with any distribution having
Seven-Segment LED Displays
Seven-Segment LED Displays Nicholas Neumann 11/19/2010 Abstract Seven-segment displays are electronic display devices used as an easy way to display decimal numerals and an alterative to the more complex
Lecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
The last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: -Our standard number system is base, also
COMBINATIONAL CIRCUITS
COMBINATIONAL CIRCUITS http://www.tutorialspoint.com/computer_logical_organization/combinational_circuits.htm Copyright tutorialspoint.com Combinational circuit is a circuit in which we combine the different
Count the Dots Binary Numbers
Activity 1 Count the Dots Binary Numbers Summary Data in computers is stored and transmitted as a series of zeros and ones. How can we represent words and numbers using just these two symbols? Curriculum
Chapter 5. Selection 5-1
Chapter 5 Selection 5-1 Selection (Decision) The second control logic structure is selection: Selection Choosing between two or more alternative actions. Selection statements alter the sequential flow
Section 1.5 Exponents, Square Roots, and the Order of Operations
Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.
Let s put together a Manual Processor
Lecture 14 Let s put together a Manual Processor Hardware Lecture 14 Slide 1 The processor Inside every computer there is at least one processor which can take an instruction, some operands and produce
An Integer Square Root Algorithm
An Integer Square Root Algorithm 71 Example 24 An Integer Square Root Algorithm The C algorithm shown in Fig. 2.8 performs an integer square root of the input a as shown in Table 2.1. Note from Table 2.1
Two-level logic using NAND gates
CSE140: Components and Design Techniques for Digital Systems Two and Multilevel logic implementation Tajana Simunic Rosing 1 Two-level logic using NND gates Replace minterm ND gates with NND gates Place
CS 3719 (Theory of Computation and Algorithms) Lecture 4
CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a
CPSC 121: Models of Computation Assignment #4, due Wednesday, July 22nd, 2009 at 14:00
CPSC 2: Models of Computation ssignment #4, due Wednesday, July 22nd, 29 at 4: Submission Instructions Type or write your assignment on clean sheets of paper with question numbers prominently labeled.
Student Exploration: Circuits
Name: Date: Student Exploration: Circuits Vocabulary: ammeter, circuit, current, ohmmeter, Ohm s law, parallel circuit, resistance, resistor, series circuit, voltage Prior Knowledge Questions (Do these
1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables