Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007)


 Isabella Violet McKenzie
 1 years ago
 Views:
Transcription
1 Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007) Problem 1: Specify two different predicates P (x) and Q(x) over the set of positive integers so that the proposition x (P (x) Q(x)) is false while x (P (x) Q(x)) is true. Your answer should contain a clear definition of the predicates P (x) and Q(x) and an explanation of why, for your choice of predicates, x (P (x) Q(x)) is false and x (P (x) Q(x)) is true. Let P (x) denote the predicate x is divisible by 3 and let Q(x) denote the predicate x is divisible by 7. The universe for both the predicates is the set of positive integers. Here, the proposition x (P (x) Q(x)) is false. The reason is that an integer x need not be divisible by both 3 and 7. (For example, 3 is not divisible by 7.) However, the proposition x (P (x) Q(x)) is true since we can choose x = 21 which is divisible by both 3 and 7. Problem 2: Suppose p, q and r are propositions such that p q, ( p) r and r (p q) are all true. Show that q is true. We will use a proof by contradiction. So, suppose q is false. (1) Note that p q is true; that is, p q is true. Since q is false, we must conclude that p is true; that is, p is false. (2) Since p is true (from (1)) and ( p) r is true (given), r is true (by modus ponens). (3) Since r is true (from (2)) and r (p q) is true (given), p q is true (by modus ponens). (4) However, p is false (from (1)) and q is false by our assumption. So, p q is false. This contradicts the conclusion in (3) that p q is true. Thus, q must be true. Problem 3: Recall that a binary propositional operator is associative if for all propositions a, b and c, the propositional forms a (b c) and (a b) c have the same truth value. Also recall that the NAND operator ( ) is defined by a b (a b). Is the NAND operator associative? Justify your answer. The NAND operator is not associative. To see this, consider the case where a = b = 1 and c = 0. The truth values of a (b c) and (a b) c are computed below. (i) a (b c) 1 (1 0) (ii) (a b) c (1 1) Thus, a (b c) and (a b) c have different truth values. So, NAND is not associative. 1
2 Problem 4: Suppose A and B are sets such that A = 6, B = 2 and A B = 1. Calculate P(A B). (Recall that for a set X, P(X) denotes the powerset of X.) By the inclusionexclusion formula, A B = A + B A B = = 7. Recall that for any finite set X, P(X) = 2 X. Therefore, P(A B) = 2 7 = 128. Problem 5: Let A = {1, 2, 3, 4} and B = {a, b, c, d, e}. How many functions from A to B are either onetoone or map the element 1 to c? (You need not simplify your answer.) Let S 1 denote the set of functions from A to B which are onetoone and let S 2 denote the set of functions from A to B which map the element 1 to c. The required answer to the problem is S 1 S 2. We will compute this number using the inclusionexclusion formula: S 1 S 2 = S 1 + S 2 S 1 S 2. (a) S 1 is the number of onetoone functions from A to B. In constructing such a function, there are 5 choices for the element 1, 4 choices for the element 2, 3 choices for the element 3 and 2 choices for the element 4. So, S 1 = = 120. (b) S 2 is the number of functions from A to B that map the element 1 to c. In constructing such a function, there is only one choice for the element 1 but there are 5 choices for each of the three remaining elements of A. So, S 2 = = 5 3 = 125. (c) S 1 S 2 is the number of functions from A to B which are onetoone and which map the element 1 to c. In constructing such a function, there are is only one choice for the element 1, 4 choices for the element 2, 3 choices for the element 3 and 2 choices for the element 4. So, S 1 S 2 = = 24. Thus, using the inclusionexclusion formula, the required answer is = 221. Problem 6: Let X = {a, b, c, d, e}. Let us call a binary relation R on X special if it satisfies all of the following conditions: (i) R is reflexive, (ii) R is symmetric and (iii) R contains the pair (a, b). Find the number of special binary relations on X. You need not simplify your answer. Every binary relation on X can be represented by a 5 5 Boolean matrix. There are 25 entries in this matrix, with 5 entries along the main diagonal, 10 entries above the diagonal and 10 entries below the diagonal. Since the relation is required to be reflexive, there is only one choice (namely, the value 1) for each entry along the diagonal. Since the relation is required to be symmetric, once we choose a 0 or 1 value for each of the 10 entries above the diagonal, the 10 entries below the diagonal are determined; that is, there is only one choice for each of those entries. Of the 10 entries above the diagonal, one entry corresponding to (a, b) must be 1 (since the relation must contain that pair). Thus, only 9 entries in the Boolean matrix have two choices each. All other entries have only one choice. So, the number of special relations = 2 9. (This expression can be simplified to 512.) 2
3 Problem 7: Let Y = {1, 2, 3, 4}. Consider the binary relation R on Y defined by Find the transitive closure of R. R = {(1, 2), (2, 1), (2, 3), (2, 4), (4, 3), (4, 4)}. The directed graph of the binary relation R is shown below Recall that the transitive closure of R contains each pair (x, y) such that there is a directed path of length 1 from x to y in the above graph. Thus, the transitive closure of R is given by {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (4, 3), (4, 4)}. Problem 8: Let N + denote the set of positive integers (i.e., integers that are strictly greater than 0). Consider the function f from N + to N + defined as follows: f(x) = 1+ the number of 9 s in the decimal representation of x. For example, f(1) = 1, f(293) = 2, f(1929) = Is f onetoone? Justify your answer. 2. Is f onto? Justify your answer. Part (a): f is not onetoone. To see this, notice that f(1) = f(2) = 1; that is, two different elements of N + are mapped by f to the same element of N +. So, f is not onetoone. Part (b): f is onto. To prove this statement, we must show that for any integer y N +, there is an integer x N + such that f(x) = y. There are two cases. Case 1: y = 1. For this case, let x = 1. From the definition of f, we have f(x) = f(1) = 1 = y. Case 2: y 2. For this case, let x be the decimal integer that consists of y 1 occurrences of the digit 9. (For example, if y = 3, we let x = 99.) Since y 2, this integer is well defined; it has at least one 9. Again, by the definition of f, f(x) = 1 + y 1 = y. Thus, for any integer y N +, there is an integer x N + such that f(x) = y. In other words, f is onto. 3
4 Problem 9: Let R denote the set of real numbers. Consider the function f from R R to R R defined by f(x, y) = (x + y, x y). Find f 1. (You may assume without proof that f is a bijection.) For any (a, b) R R, let f 1 (a, b) = (x, y). Thus, to specify f 1, we need to do the following: given a and b, find the values of x and y. By the definition of inverse, we have f(x, y) = (a, b). However, by the definition of f, we have f(x, y) = (x + y, x y). Therefore, we have two linear equations x + y = a and x y = b for the variables x and y. Solving the two linear equations, we get In other words, f 1 is defined by x = (a + b)/2 and y = (a b)/2. f 1 (a, b) = ( (a + b)/2, (a b)/2 ). Problem 10: Find the number of solutions to the equation x 1 + x 2 + x 3 + x 4 + x 5 = 37, where x 1, x 2, x 3, x 4 and x 5 are nonnegative integers, x 2 8, x 3 7, x 4 2 and x 5 < 4. (You may leave the answer as an expression consisting of binomial coefficients.) To solve this problem, we will use the fact that the number of solutions to the equation z 1 + z z r = q where z 1, z 2,..., z r and q are all nonnegative integers, is = C(q + r 1, r 1). To handle the constraints on x 2, x 3 and x 4, first define new variables y 2 = x 2 8, y 3 = x 3 7 and y 4 = x 4 2. Since x 2 8, x 3 7 and x 4 2, we have y 2 0, y 3 0 and y 4 0. Substituting for x 2, x 3 and x 4 in terms of y 2, y 3 and y 4 respectively in the given equation, we get x 1 + y 2 + y 3 + y 4 + x 5 = = 20 (1) Let N 1 denote the number of solutions to Equation (1) where each of the variables is a nonnegative integer. Using the formula mentioned above, N 1 = C( , 5 1) = C(24, 4). In some of these solutions, x 5 satisfies the condition x 5 < 4 while in others, x 5 4. So, if we find the number of solutions, say N 2, where x 5 4, then the required solution to the problem is N 1 N 2. To find N 2, we proceed in a manner similar to that of N 1. Define a new variable y 5 = x 5 4. Thus, when x 5 4, y 5 0. We substitute for x 5 in terms of y 5 in Equation (1) to get the equation x 1 + y 2 + y 3 + y 4 + y 5 = 20 4 = 16 (2) The number of solutions N 2 to Equation (2) where each variable is a nonnegative integer is = C( , 5 1) = C(20, 4). Therefore, the required answer is N 1 N 2 = C(24, 4) C(20, 4). 4
5 Problem 11: Recall that a bit string is a string composed of characters 0 and 1. Let us call a bit string s interesting if it satisfies all of the following conditions: (i) s has length 23, (ii) s starts with 1110, (iii) s ends with with and (iv) s has 0 as its middle bit. Find the number of bit strings that are interesting. (You need not simplify your answer.) There are 23 positions in the string. Of these, 10 positions (namely, the first 4, the last 5 and the middle bit) have specified values. For each of the remaining 13 positions, we have two choices (namely, 0 or 1). So, the number of interesting bit strings = (This expression can be simplified to 8192.) Problem 12: Use induction on n to prove the following for all n 2: If A 1, A 2,..., A n are subsets of a universal set U, then A 1 A 2... A n = A 1 A 2... A n. You can assume without proof that for any two subsets X and Y of the universal set U, X Y = X Y. Proof: Basis: n = 2. For any two sets A 1 and A 2, the problem allows us to assume that A 1 A 2 = A 1 A 2. Thus, the basis holds. Induction Hypothesis: A 1, A 2,..., A k, Assume that for some integer k 2, and any collection of k sets A 1 A 2... A k = A 1 A 2... A k (3) To prove: For any collection of k + 1 sets A 1, A 2,..., A k+1, A 1 A 2... A k A k+1 = A 1 A 2... A k A k+1 (4) Proof: Consider any collection of k + 1 sets A 1, A 2,..., A k+1. Let X = A 1 A 2... A k. Consider the two sets X and A k+1. By the given assumption for two sets, we have X A k+1 = X A k+1 (5) Now, by the definition of X, X = A 1 A 2... A k. Therefore, we can apply the inductive hypothesis (Equation (3)) to X to get X = A 1 A 2... A k (6) Substituting the result of Equation (6) in Equation (5), we get Again, using the definition of X in Equation (7), we get as required. This completes the proof. X A k+1 = A 1 A 2... A k A k+1 (7) A 1 A 2... A k A k+1 = A 1 A 2... A k A k+1 (8) 5
Packet #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 313 Course Objectives At the conclusion of this course, you should
More informationMAT 243 Test 1 SOLUTIONS, FORM A
t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,
More informationADVANCED CALCULUS  MTH433 LECTURE 4  FINITE AND INFINITE SETS
ADVANCED CALCULUS  MTH433 LECTURE 4  FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Copyright Cengage Learning. All rights reserved. SECTION 5.4 Strong Mathematical Induction and the WellOrdering Principle for the Integers Copyright
More informationCPSC 121 Sample Final Examination December 2013
CPSC 121 Sample Final Examination December 201 [6] 1. Short answers [] a. What is wrong with the following circuit? You can not connect the outputs of two or more gates together directly; what will happen
More informationCSCE 222 Discrete Structures for Computing. Predicate Logic. Dr. Hyunyoung Lee. !!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Predicate Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Predicates A function P from a set D to the set Prop of propositions is called a predicate.
More informationINFINITY: CARDINAL NUMBERS
INFINITY: CARDINAL NUMBERS BJORN POONEN 1 Some terminology of set theory N := {0, 1, 2, 3, } Z := {, 2, 1, 0, 1, 2, } Q := the set of rational numbers R := the set of real numbers C := the set of complex
More informationMath 230 Final Exam, Spring 2008
c IIT Dept. Applied Mathematics, May 15, 2008 1 PRINT Last name: Signature: First name: Student ID: Math 230 Final Exam, Spring 2008 Conditions. 2 hours. No book, notes, calculator, cell phones, etc. Part
More informationPropositional Logic, Predicates, and Equivalence
Chapter 1 Propositional Logic, Predicates, and Equivalence A statement or a proposition is a sentence that is true (T) or false (F) but not both. The symbol denotes not, denotes and, and denotes or. If
More informationCSE 1400 Applied Discrete Mathematics Proofs
CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4
More informationMATH 13 SAMPLE FINAL EXAM SOLUTIONS
MATH 13 SAMPLE FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers.
More informationChapter 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
More informationMathematical Preliminaries. Sipser pages 128
Mathematical Preliminaries Sipser pages 128 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation
More informationDISCRETE MATH: FINAL REVIEW
DISCRETE MATH: FINAL REVIEW DR. DANIEL FREEMAN 1) a. Does 3 = {3}? b. Is 3 {3}? c. Is 3 {3}? c. Is {3} {3}? c. Is {3} {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z x > 0} {x R x > 0}? 1. Chapter 1 review
More informationSets are one of the basic building blocks for the types of objects considered in discrete mathematics.
Section 2.1 Introduction Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Important for counting. Programming languages have set operations. Set theory
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationSolutions to Homework Set 1
Solutions to Homework Set 1 1. Prove that notq notp implies P Q. In class we proved that A B implies notb nota Replacing the statement A by the statement notq and the statement B by the statement
More informationSupplementary Material for MTH 299 Online Edition
Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think
More informationCS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:
x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which
More informationDiscrete Structures CRN Test 3 Version 1 CMSC 2123 Autumn 2013
. Print your name on your scantron in the space labeled NAME. 2. Print CMSC 223 in the space labeled SUBJECT. 3. Print the date 22203, in the space labeled DATE. 4. Print your CRN, 786, in the space
More informationMATH 201 Solutions: TEST 3A (in class)
MATH 201 Solutions: TEST 3A (in class) (revised) God created infinity, and man, unable to understand infinity, had to invent finite sets.  Gian Carlo Rota Part I [5 pts each] 1. Let X be a set. Define
More informationClimbing an Infinite Ladder
Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite
More informationThe proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence.
The Conditional (IMPLIES) Operator The conditional operation is written p q. The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence. The Conditional
More information6 CARDINALITY OF SETS
6 CARDINALITY OF SETS MATH10111  Foundations of Pure Mathematics We all have an idea of what it means to count a finite collection of objects, but we must be careful to define rigorously what it means
More information2 Truth Tables, Equivalences and the Contrapositive
2 Truth Tables, Equivalences and the Contrapositive 12 2 Truth Tables, Equivalences and the Contrapositive 2.1 Truth Tables In a mathematical system, true and false statements are the statements of the
More information5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.
Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent
More informationA statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
More informationMATH 114 Fall 2004 Solutions to practice problems for Final Exam
MATH 11 Fall 00 Solutions to practice problems for Final Exam Reminder: the final exam is on Monday, December 13 from 11am  1am. Office hours: Thursday, December 9 from 15pm; Friday, December 10 from
More informationCSE 20. Lecture 4: Introduction to Boolean algebra. CSE 20: Lecture4
CSE 20 Lecture 4: Introduction to Boolean algebra Reminder First quiz will be on Friday (17th January) in class. It is a paper quiz. Syllabus is all that has been done till Wednesday. If you want you may
More informationMathematical Induction
Mathematical Induction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Mathematical Induction Fall 2014 1 / 21 Outline 1 Mathematical Induction 2 Strong Mathematical
More informationWUCT121. Discrete Mathematics. Logic. Tutorial Exercises
WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the
More informationA Semester Course in Basic Abstract Algebra
A Semester Course in Basic Abstract Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved December 29, 2011 1 PREFACE This book is an introduction to abstract algebra course for undergraduates
More information2. Polynomials. 19 points. 3/3/3/3/3/4 Clearly indicate your correctly formatted answer: this is what is to be graded. No need to justify!
1. Short Modular Arithmetic/RSA. 16 points: 3/3/3/3/4 For each question, please answer in the correct format. When an expression is asked for, it may simply be a number, or an expression involving variables
More informationGroup 4 Project. James Earthman, Gregg Greenleaf, Zach Harvey, Zac Leids
Group 4 Project James Earthman, Gregg Greenleaf, Zach Harvey, Zac Leids 2.5 #21 21. Give the output signals S and T for the circuit in the right column if the input signals P, Q, and R are as specified.
More informationDiscrete Mathematical Structures: Theory and Applications
Chapter 1: Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications Learning Objectives Learn about sets Explore various operations on sets Become familiar with
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationn n P} is a bounded subset Proof. Let A be a nonempty subset of Z, bounded above. Define the set
1 Mathematical Induction We assume that the set Z of integers are well defined, and we are familiar with the addition, subtraction, multiplication, and division. In particular, we assume the following
More informationLogical Reasoning. Chapter Statements and Logical Operators
Chapter 2 Logical Reasoning 2.1 Statements and Logical Operators Preview Activity 1 (Compound Statements) Mathematicians often develop ways to construct new mathematical objects from existing mathematical
More informationProof by Contradiction
Proof by Contradiction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Proof by Contradiction Fall 2014 1 / 12 Outline 1 Proving Statements with Contradiction 2 Proving
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20ab/ Today's learning goals Define and compute the cardinality of a set. Use functions to compare the sizes of sets. Classify sets
More informationBoolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.
The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or
More informationCS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)
CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bioconditional Converse Inverse Contrapositive Laws of
More informationLogic. Facts (with proofs) CHAPTER 1. Definitions
CHAPTER 1 Logic Definitions D1. Statements (propositions), compound statements. D2. Truth values for compound statements p q, p q, p q, p q. Truth tables. D3. Converse and contrapositive. D4. Tautologies
More informationDefinition 2. Conjunction of p and q
Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football
More informationCS Discrete Mathematics Dr. D. Manivannan (Mani)
CS 275  Discrete Mathematics Dr. D. Manivannan (Mani) Department of Computer Science University of Kentucky Lexington, KY 40506 Course Website: www.cs.uky.edu/~manivann/cs275 Notes based on Discrete Mathematics
More informationCHAPTER 1. MATHEMATICAL LOGIC 1.1 Fundamentals of Mathematical Logic
CHAPER 1 MAHEMAICAL LOGIC 1.1 undamentals of Mathematical Logic Logic is commonly known as the science of reasoning. Some of the reasons to study logic are the following: At the hardware level the design
More informationA. Propositional Logic
CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals
More informationWeek 45: Binary Relations
1 Binary Relations Week 45: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationCSE 321 Solutions to Practice Problems
CSE 321 Solutions to Practice Problems Instructions: Feel free NOT to multiply out binomial coefficients, factorials, etc, and feel free to leave answers in the form of a sum. No calculators, books or
More informationArgument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday.
Logic and Proof Argument An argument is a sequence of statements. All statements but the first one are called assumptions or hypothesis. The final statement is called the conclusion. An argument is valid
More informationDiscrete Structures Homework 1
Discrete Structures Homework 1 Due: June 15. Section 1.1 16 Determine whether these biconditionals are true or false. a) 2 + 2 = 4 if and only if 1 + 1 = 2 b) 1 + 1 = 2 if and only if 2 + 3 = 4 c) 1 +
More informationnot to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results
REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division
More informationPropositional Logic: BottomUp Proofs
Propositional Logic: BottomUp Proofs CPSC 322 Logic 3 Textbook 5.2 Propositional Logic: BottomUp Proofs CPSC 322 Logic 3, Slide 1 Lecture Overview 1 Recap 2 BottomUp Proofs 3 Soundness of BottomUp
More informationPropositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173
Propositional Logic CSC 17 Propositional logic mathematical model (or algebra) for reasoning about the truth of logical expressions (propositions) Logical expressions propositional variables or logical
More informationCombinational Logic. By : Ali Mustafa
Combinational Logic By : Ali Mustafa Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer How to Analyze any combinational circuit like this? Analysis Procedure To obtain the output
More informationTopics in Logic and Proofs
Chapter 2 Topics in Logic and Proofs Some mathematical statements carry a logical value of being true or false, while some do not. For example, the statement 4 + 5 = 9 is true, whereas the statement 2
More informationModal Logic. UIT2206: The Importance of Being Formal. Martin Henz. March 19, 2014
Modal Logic UIT2206: The Importance of Being Formal Martin Henz March 19, 2014 1 Motivation The source of meaning of formulas in the previous chapters were models. Once a particular model is chosen, say
More informationSet Theory. CSE 215, Foundations of Computer Science Stony Brook University
Set Theory CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Set theory Abstract set theory is one of the foundations of mathematical thought Most mathematical
More informationFirst order Logic ( Predicate Logic) and Methods of Proof
First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating
More informationA Short Review of Cardinality
Christopher Heil A Short Review of Cardinality November 14, 2017 c 2017 Christopher Heil Chapter 1 Cardinality We will give a short review of the definition of cardinality and prove some facts about the
More informationD I S C R E T E M AT H E M AT I C S H O M E W O R K
D E PA R T M E N T O F C O M P U T E R S C I E N C E S C O L L E G E O F E N G I N E E R I N G F L O R I D A T E C H D I S C R E T E M AT H E M AT I C S H O M E W O R K W I L L I A M S H O A F F S P R
More informationPropositional Logic. Fall () Propositional Logic Fall / 30
Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically
More informationANS: If you are in Kwangju then you are in South Korea but not in Seoul.
Math 15  Spring 2017  Homework 1.1 and 1.2 Solutions 1. (1.1#1) Let the following statements be given. p = There is water in the cylinders. q = The head gasket is blown. r = The car will start. (a) Translate
More informationBasic Proof Examples
Basic Proof Examples Lisa Oberbroeckling Loyola University Maryland Fall 2015 Note. In this document, we use the symbol as the negation symbol. Thus p means not p. There are four basic proof techniques
More informationSection 7.5: Cardinality
Section 7: Cardinality In this section, we shall consider extend some of the ideas we have developed to infinite sets One interesting consequence of this discussion is that we shall see there are as many
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationChapter Summary. Mathematical Induction Strong Induction WellOrdering Recursive Definitions Structural Induction Recursive Algorithms
1 Chapter Summary Mathematical Induction Strong Induction WellOrdering Recursive Definitions Structural Induction Recursive Algorithms 2 Section 5.1 3 Section Summary Mathematical Induction Examples of
More informationProofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction
Introduction I Proofs Computer Science & Engineering 235 Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu A proof is a proof. What kind of a proof? It s a proof. A proof is a proof. And when
More informationLOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D.
Basic Eperiment and Design of Electronics LOGIC GATES Ho Kyung Kim, Ph.D. hokyung@pusan.ac.kr School of Mechanical Engineering Pusan National University Outline Boolean algebra Logic gates Karnaugh maps
More informationWe introduce one more operation on sets, perhaps the most important
11. The power set Please accept my resignation. I don t want to belong to any club that will accept me as a member. Groucho Marx We introduce one more operation on sets, perhaps the most important one:
More informationMathematical Induction
Chapter 6 Mathematical Induction 6.1 The Process of Mathematical Induction 6.1.1 Motivating Mathematical Induction Consider the sum of the first several odd integers. produce the following: 1 = 1 1 + 3
More information447 HOMEWORK SET 1 IAN FRANCIS
7 HOMEWORK SET 1 IAN FRANCIS For each n N, let A n {(n 1)k : k N}. 1 (a) Determine the truth value of the statement: for all n N, A n N. Justify. This statement is false. Simply note that for 1 N, A 1
More informationMaanavaN.Com MA1256 DISCRETE MATHEMATICS. DEPARTMENT OF MATHEMATICS QUESTION BANK Subject & Code : MA1256 DISCRETE MATHEMATICS
DEPARTMENT OF MATHEMATICS QUESTION BANK Subject & Code : UNIT I PROPOSITIONAL CALCULUS Part A ( Marks) Year / Sem : III / V. Write the negation of the following proposition. To enter into the country you
More informationEQUIVALENCE RELATIONS (NOTES FOR STUDENTS) 1. RELATIONS
EQUIVALENCE RELATIONS (NOTES FOR STUDENTS) LIOR SILBERMAN Version 1.0 compiled September 9, 2015. 1.1. List of examples. 1. RELATIONS Equality of real numbers: for some x,y R we have x = y. For other pairs
More informationProof. Theorems. Theorems. Example. Example. Example. Part 4. The Big Bang Theory
Proof Theorems Part 4 The Big Bang Theory Theorems A theorem is a statement we intend to prove using existing known facts (called axioms or lemmas) Used extensively in all mathematical proofs which should
More informationMathematical Induction. How does discrete math help us. How does discrete math help (CS160)? How does discrete math help (CS161)?
How does discrete math help us Helps create a solution (program) Helps analyze a program How does discrete math help (CS160)? Helps create a solution (program) q Logic helps you understand conditionals
More informationCS 154, Lecture 4: Limitations on DFAs (I), Pumping Lemma, Minimizing DFAs
CS 154, Lecture 4: Limitations on FAs (I), Pumping Lemma, Minimizing FAs Regular or Not? NonRegular Languages = { w w has equal number of occurrences of 01 and 10 } REGULAR! C = { w w has equal number
More informationLogic: The Big Picture
Logic: The Big Picture A typical logic is described in terms of syntax: what are the legitimate formulas semantics: under what circumstances is a formula true proof theory/ axiomatization: rules for proving
More informationBoolean Algebra and Digital Logic
All modern digital computers are dependent on circuits that implement Boolean functions. We shall discuss two classes of such circuits: Combinational and Sequential. The difference between the two types
More informationAdvanced Topics in LP and FP
Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection
More informationUniversity of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura. February 9, :30 pm Duration: 1:50 hs. Closed book, no calculators
University of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura February 9, 2010 11:30 pm Duration: 1:50 hs Closed book, no calculators Last name: First name: Student number: There are 5 questions and
More information10.4 The Kruskal Katona theorem
104 The Krusal Katona theorem 141 Example 1013 (Maximum weight traveling salesman problem We are given a complete directed graph with nonnegative weights on edges, and we must find a maximum weight Hamiltonian
More informationChapter 1. Logic and Proof
Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known
More informationMath 13, Spring 2013, Lecture B: Midterm
Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # Email address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.
More information1. Propositions: Contrapositives and Converses
Preliminaries 1 1. Propositions: Contrapositives and Converses Given two propositions P and Q, the statement If P, then Q is interpreted as the statement that if the proposition P is true, then the statement
More informationHANDOUT AND SET THEORY. Ariyadi Wijaya
HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics
More informationNotes on induction proofs and recursive definitions
Notes on induction proofs and recursive definitions James Aspnes December 13, 2010 1 Simple induction Most of the proof techniques we ve talked about so far are only really useful for proving a property
More informationDiscrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
More informationPropositional Logics and their Algebraic Equivalents
Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic
More information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
More informationcse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska
cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 1 Course Web Page www3.cs.stonybrook.edu/ cse303 The webpage contains: lectures notes slides; very detailed solutions to
More informationSection 2.1: Introduction to the Logic of Quantified Statements
Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional
More informationMathematical Induction. Rosen Chapter 4.1,4.2 (6 th edition) Rosen Ch. 5.1, 5.2 (7 th edition)
Mathematical Induction Rosen Chapter 4.1,4.2 (6 th edition) Rosen Ch. 5.1, 5.2 (7 th edition) Motivation Suppose we want to prove that for every value of n: 1 + 2 + + n = n(n + 1)/2. Let P(n) be the predicate
More informationMATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.
MATH 4 Summer 011 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. WellOrdering Axiom for the Integers If
More informationChapter 5: The Integers
c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition
More informationLogic. Propositional Logic: Syntax
Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about
More informationRelations. Binary Relation. Let A and B be sets. A (binary) relation from A to B is a subset of A B. Notation. Let R A B be a relation from A to B.
Relations Binary Relation Let A and B be sets. A (binary) relation from A to B is a subset of A B. Notation Let R A B be a relation from A to B. If (a, b) R, we write a R b. 1 Binary Relation Example:
More informationCh 7 Summary  POLYNOMIAL FUNCTIONS
Ch 7 Summary  POLYNOMIAL FUNCTIONS 1. An opentop box is to be made by cutting congruent squares of side length x from the corners of a 8.5 by 11inch sheet of cardboard and bending up the sides. a)
More informationAdvanced Algebra. MA Prof. Götz Pfeiffer
Advanced. Prof. Götz Pfeiffer http://schmidt.nuigalway.ie/ma1804 School of Mathematics, Statistics and Applied Mathematics NUI Galway Semester 2 (2017/2018) Outline 1 2 3 References. Norman L. Biggs.
More informationEconomics 204 Fall 2011 Problem Set 1 Suggested Solutions
Economics 204 Fall 2011 Problem Set 1 Suggested Solutions 1. Suppose k is a positive integer. Use induction to prove the following two statements. (a) For all n N 0, the inequality (k 2 + n)! k 2n holds.
More information