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

Size: px
Start display at page:

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

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 inclusion-exclusion 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 one-to-one 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 one-to-one 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 inclusion-exclusion formula: S 1 S 2 = S 1 + S 2 S 1 S 2. (a) S 1 is the number of one-to-one 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 one-to-one 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 inclusion-exclusion 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 one-to-one? Justify your answer. 2. Is f onto? Justify your answer. Part (a): f is not one-to-one. 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 one-to-one. 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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

Problem 1: Suppose A, B, C and D are finite sets such that A B = C D and C = D. Prove or disprove: A = B.

Problem 1: Suppose A, B, C and D are finite sets such that A B = C D and C = D. Prove or disprove: A = B. Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination III (Spring 2007) Problem 1: Suppose A, B, C and D are finite sets

More information

Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination I (Spring 2008)

Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination I (Spring 2008) Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination I (Spring 2008) Problem 1: Suppose A, B, C and D are arbitrary sets.

More information

Packet #1: Logic & Proofs. Applied Discrete Mathematics

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 3-13 Course Objectives At the conclusion of this course, you should

More information

1. Consider the conditional E = p q r. Use de Morgan s laws to write simplified versions of the following : The negation of E : 5 points

1. Consider the conditional E = p q r. Use de Morgan s laws to write simplified versions of the following : The negation of E : 5 points Introduction to Discrete Mathematics 3450:208 Test 1 1. Consider the conditional E = p q r. Use de Morgan s laws to write simplified versions of the following : The negation of E : The inverse of E : The

More information

Conjunction: p q is true if both p, q are true, and false if at least one of p, q is false. The truth table for conjunction is as follows.

Conjunction: p q is true if both p, q are true, and false if at least one of p, q is false. The truth table for conjunction is as follows. Chapter 1 Logic 1.1 Introduction and Definitions Definitions. A sentence (statement, proposition) is an utterance (that is, a string of characters) which is either true (T) or false (F). A predicate is

More information

Predicate Logic. Andreas Klappenecker

Predicate Logic. Andreas Klappenecker Predicate Logic Andreas Klappenecker Predicates A function P from a set D to the set Prop of propositions is called a predicate. The set D is called the domain of P. Example Let D=Z be the set of integers.

More information

Exam Practice Problems

Exam Practice Problems Math 231 Exam Practice Problems WARNING: This is not a sample test. Problems on the exams may or may not be similar to these problems. These problems are just intended to focus your study of the topics.

More information

Packet #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics

Packet #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus

More information

Logic, Sets, and Proofs

Logic, Sets, and Proofs Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.

More information

Name (please print) Mathematics Final Examination December 14, 2005 I. (4)

Name (please print) Mathematics Final Examination December 14, 2005 I. (4) Mathematics 513-00 Final Examination December 14, 005 I Use a direct argument to prove the following implication: The product of two odd integers is odd Let m and n be two odd integers Since they are odd,

More information

(1) Which of the following are propositions? If it is a proposition, determine its truth value: A propositional function, but not a proposition.

(1) Which of the following are propositions? If it is a proposition, determine its truth value: A propositional function, but not a proposition. Math 231 Exam Practice Problem Solutions WARNING: This is not a sample test. Problems on the exams may or may not be similar to these problems. These problems are just intended to focus your study of the

More information

At least one of us is a knave. What are A and B?

At least one of us is a knave. What are A and B? 1. This is a puzzle about an island in which everyone is either a knight or a knave. Knights always tell the truth and knaves always lie. This problem is about two people A and B, each of whom is either

More information

ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS

ADVANCED 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 information

CSE 1400 Applied Discrete Mathematics Definitions

CSE 1400 Applied Discrete Mathematics Definitions CSE 1400 Applied Discrete Mathematics Definitions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Arithmetic 1 Alphabets, Strings, Languages, & Words 2 Number Systems 3 Machine

More information

Lecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel

Lecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical

More information

MAT 243 Test 1 SOLUTIONS, FORM A

MAT 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 information

MATH 3300 Test 1. Name: Student Id:

MATH 3300 Test 1. Name: Student Id: Name: Student Id: There are nine problems (check that you have 9 pages). Solutions are expected to be short. In the case of proofs, one or two short paragraphs should be the average length. Write your

More information

Math 564 Homework 1. Solutions.

Math 564 Homework 1. Solutions. Math 564 Homework 1. Solutions. Problem 1. Prove Proposition 0.2.2. A guide to this problem: start with the open set S = (a, b), for example. First assume that a >, and show that the number a has the properties

More information

Discrete Structures: Sample Questions, Exam 2, SOLUTIONS

Discrete Structures: Sample Questions, Exam 2, SOLUTIONS Discrete Structures: Sample Questions, Exam 2, SOLUTIONS (This is longer than the actual test.) 1. Show that any postage of 8 cents or more can be achieved by using only -cent and 5-cent stamps. We proceed

More information

COMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University

COMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called

More information

CS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Midterm 1 Solutions

CS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Midterm 1 Solutions CS 70 Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Midterm 1 Solutions PRINT Your Name: Answer: Oski Bear SIGN Your Name: PRINT Your Student ID: CIRCLE your exam room: Dwinelle

More information

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k)

(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k) MATH 111 Optional Exam 3 lutions 1. (0 pts) We define a relation on Z as follows: a b if a + b is divisible by 3. (a) (1 pts) Prove that is an equivalence relation. (b) (8 pts) Determine all equivalence

More information

SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION

SEQUENCES, 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 Well-Ordering Principle for the Integers Copyright

More information

Math 10850, fall 2017, University of Notre Dame

Math 10850, fall 2017, University of Notre Dame Math 10850, fall 2017, University of Notre Dame Notes on first exam September 22, 2017 The key facts The first midterm will be on Thursday, September 28, 6.15pm-7.45pm in Hayes-Healy 127. What you need

More information

F 2k 1 = F 2n. for all positive integers n.

F 2k 1 = F 2n. for all positive integers n. Question 1 (Fibonacci Identity, 15 points). Recall that the Fibonacci numbers are defined by F 1 = F 2 = 1 and F n+2 = F n+1 + F n for all n 0. Prove that for all positive integers n. n F 2k 1 = F 2n We

More information

Practice Exam 1 CIS/CSE 607, Spring 2009

Practice Exam 1 CIS/CSE 607, Spring 2009 Practice Exam 1 CIS/CSE 607, Spring 2009 Problem 1) Let R be a reflexive binary relation on a set A. Prove that R is transitive if, and only if, R = R R. Problem 2) Give an example of a transitive binary

More information

cse 311: foundations of computing Fall 2015 Lecture 6: Predicate Logic, Logical Inference

cse 311: foundations of computing Fall 2015 Lecture 6: Predicate Logic, Logical Inference cse 311: foundations of computing Fall 2015 Lecture 6: Predicate Logic, Logical Inference quantifiers x P(x) P(x) is true for every x in the domain read as for all x, P of x x P x There is an x in the

More information

CPSC 121 Sample Final Examination December 2013

CPSC 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 information

INFINITY: CARDINAL NUMBERS

INFINITY: 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 information

Discrete Mathematics Fall 2018 Midterm Exam Prof. Callahan. Section: NetID: Multiple Choice Question (30 questions in total, 4 points each)

Discrete Mathematics Fall 2018 Midterm Exam Prof. Callahan. Section: NetID: Multiple Choice Question (30 questions in total, 4 points each) Discrete Mathematics Fall 2018 Midterm Exam Prof. Callahan Section: NetID: Name: Multiple Choice Question (30 questions in total, 4 points each) 1 Consider the following propositions: f: The student got

More information

Math 230 Final Exam, Spring 2008

Math 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 information

MATH 363: Discrete Mathematics

MATH 363: Discrete Mathematics MATH 363: Discrete Mathematics Learning Objectives by topic The levels of learning for this class are classified as follows. 1. Basic Knowledge: To recall and memorize - Assess by direct questions. The

More information

6 Permutations Very little of this section comes from PJE.

6 Permutations Very little of this section comes from PJE. 6 Permutations Very little of this section comes from PJE Definition A permutation (p147 of a set A is a bijection ρ : A A Notation If A = {a b c } and ρ is a permutation on A we can express the action

More information

Finite Automata Theory and Formal Languages TMV027/DIT321 LP Recap: Logic, Sets, Relations, Functions

Finite Automata Theory and Formal Languages TMV027/DIT321 LP Recap: Logic, Sets, Relations, Functions Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2017 Formal proofs; Simple/strong induction; Mutual induction; Inductively defined sets; Recursively defined functions. Lecture 3 Ana Bove

More information

Discrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland

Discrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................

More information

LECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel

LECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics Solution Guide

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics Solution Guide Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Solution Guide Marcel B. Finan c All Rights Reserved 2015 Edition Contents

More information

Exam in Discrete Mathematics

Exam in Discrete Mathematics Exam in Discrete Mathematics First Year at the Faculty of Engineering and Science and the Technical Faculty of IT and Design June 4th, 018, 9.00-1.00 This exam consists of 11 numbered pages with 14 problems.

More information

CSE 1400 Applied Discrete Mathematics Proofs

CSE 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 information

586 Index. vertex, 369 disjoint, 236 pairwise, 272, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

586 Index. vertex, 369 disjoint, 236 pairwise, 272, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 465 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

More information

Background for Discrete Mathematics

Background for Discrete Mathematics Background for Discrete Mathematics Huck Bennett Northwestern University These notes give a terse summary of basic notation and definitions related to three topics in discrete mathematics: logic, sets,

More information

CSCE 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 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 information

Exercises 1 - Solutions

Exercises 1 - Solutions Exercises 1 - Solutions SAV 2013 1 PL validity For each of the following propositional logic formulae determine whether it is valid or not. If it is valid prove it, otherwise give a counterexample. Note

More information

Propositional Logic, Predicates, and Equivalence

Propositional 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 information

7.11 A proof involving composition Variation in terminology... 88

7.11 A proof involving composition Variation in terminology... 88 Contents Preface xi 1 Math review 1 1.1 Some sets............................. 1 1.2 Pairs of reals........................... 3 1.3 Exponentials and logs...................... 4 1.4 Some handy functions......................

More information

Equivalence of Propositions

Equivalence of Propositions Equivalence of Propositions 1. Truth tables: two same columns 2. Sequence of logical equivalences: from one to the other using equivalence laws 1 Equivalence laws Table 6 & 7 in 1.2, some often used: Associative:

More information

CSE 20 Discrete Math. Winter, January 24 (Day 5) Number Theory. Instructor: Neil Rhodes. Proving Quantified Statements

CSE 20 Discrete Math. Winter, January 24 (Day 5) Number Theory. Instructor: Neil Rhodes. Proving Quantified Statements CSE 20 Discrete Math Proving Quantified Statements Prove universal statement: x D, P(x)Q(x) Exhaustive enumeration Generalizing from the generic particular Winter, 2006 Suppose x is in D and P(x) Therefore

More information

First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms

First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO

More information

Notation Index. gcd(a, b) (greatest common divisor) NT-16

Notation Index. gcd(a, b) (greatest common divisor) NT-16 Notation Index (for all) B A (all functions) B A = B A (all functions) SF-18 (n) k (falling factorial) SF-9 a R b (binary relation) C(n,k) = n! k! (n k)! (binomial coefficient) SF-9 n! (n factorial) SF-9

More information

Sets are one of the basic building blocks for the types of objects considered in discrete mathematics.

Sets 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 information

CIS 375 Intro to Discrete Mathematics Exam 3 (Section M004: Blue) 6 December Points Possible

CIS 375 Intro to Discrete Mathematics Exam 3 (Section M004: Blue) 6 December Points Possible Name: CIS 375 Intro to Discrete Mathematics Exam 3 (Section M004: Blue) 6 December 2016 Question Points Possible Points Received 1 12 2 14 3 14 4 12 5 16 6 16 7 16 Total 100 Instructions: 1. This exam

More information

CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati

CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) I Semester Core Course. FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) I Semester Core Course. FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) (2011 Admission Onwards) I Semester Core Course FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK 1) If A and B are two sets

More information

Math 230 Final Exam, Spring 2009

Math 230 Final Exam, Spring 2009 IIT Dept. Applied Mathematics, May 13, 2009 1 PRINT Last name: Signature: First name: Student ID: Math 230 Final Exam, Spring 2009 Conditions. 2 hours. No book, notes, calculator, cell phones, etc. Part

More information

Math 267a - Propositional Proof Complexity. Lecture #1: 14 January 2002

Math 267a - Propositional Proof Complexity. Lecture #1: 14 January 2002 Math 267a - Propositional Proof Complexity Lecture #1: 14 January 2002 Lecturer: Sam Buss Scribe Notes by: Robert Ellis 1 Introduction to Propositional Logic 1.1 Symbols and Definitions The language of

More information

MATH 13 SAMPLE FINAL EXAM SOLUTIONS

MATH 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 information

CIS 375 Intro to Discrete Mathematics Exam 3 (Section M001: Green) 6 December Points Possible

CIS 375 Intro to Discrete Mathematics Exam 3 (Section M001: Green) 6 December Points Possible Name: CIS 375 Intro to Discrete Mathematics Exam 3 (Section M001: Green) 6 December 2016 Question Points Possible Points Received 1 12 2 14 3 14 4 12 5 16 6 16 7 16 Total 100 Instructions: 1. This exam

More information

Reexam in Discrete Mathematics

Reexam in Discrete Mathematics Reexam in Discrete Mathematics First Year at the Faculty of Engineering and Science and the Technical Faculty of IT and Design August 15th, 2017, 9.00-13.00 This exam consists of 11 numbered pages with

More information

2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic

2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares

More information

CSE 20 DISCRETE MATH. Fall

CSE 20 DISCRETE MATH. Fall CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Describe and use algorithms for integer operations based on their expansions Relate algorithms for integer

More information

Economics 204 Summer/Fall 2017 Lecture 1 Monday July 17, 2017

Economics 204 Summer/Fall 2017 Lecture 1 Monday July 17, 2017 Economics 04 Summer/Fall 07 Lecture Monday July 7, 07 Section.. Methods of Proof We begin by looking at the notion of proof. What is a proof? Proof has a formal definition in mathematical logic, and a

More information

Predicate Logic: Sematics Part 1

Predicate Logic: Sematics Part 1 Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with

More information

Chapter 1 : The language of mathematics.

Chapter 1 : The language of mathematics. MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :

More information

MATH 201 Solutions: TEST 3-A (in class)

MATH 201 Solutions: TEST 3-A (in class) MATH 201 Solutions: TEST 3-A (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 information

The Process of Mathematical Proof

The Process of Mathematical Proof 1 The Process of Mathematical Proof Introduction. Mathematical proofs use the rules of logical deduction that grew out of the work of Aristotle around 350 BC. In previous courses, there was probably an

More information

3. The Logic of Quantified Statements Summary. Aaron Tan August 2017

3. The Logic of Quantified Statements Summary. Aaron Tan August 2017 3. The Logic of Quantified Statements Summary Aaron Tan 28 31 August 2017 1 3. The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I Predicate; domain; truth set Universal quantifier,

More information

Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014

Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical

More information

MATH 215 Final. M4. For all a, b in Z, a b = b a.

MATH 215 Final. M4. For all a, b in Z, a b = b a. MATH 215 Final We will assume the existence of a set Z, whose elements are called integers, along with a well-defined binary operation + on Z (called addition), a second well-defined binary operation on

More information

Meta-logic derivation rules

Meta-logic derivation rules Meta-logic derivation rules Hans Halvorson February 19, 2013 Recall that the goal of this course is to learn how to prove things about (as opposed to by means of ) classical first-order logic. So, we will

More information

Mathematical Preliminaries. Sipser pages 1-28

Mathematical Preliminaries. Sipser pages 1-28 Mathematical Preliminaries Sipser pages 1-28 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation

More information

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

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

More information

MATH FINAL EXAM REVIEW HINTS

MATH FINAL EXAM REVIEW HINTS MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any

More information

CSE 311: Foundations of Computing I Autumn 2014 Practice Final: Section X. Closed book, closed notes, no cell phones, no calculators.

CSE 311: Foundations of Computing I Autumn 2014 Practice Final: Section X. Closed book, closed notes, no cell phones, no calculators. CSE 311: Foundations of Computing I Autumn 014 Practice Final: Section X YY ZZ Name: UW ID: Instructions: Closed book, closed notes, no cell phones, no calculators. You have 110 minutes to complete the

More information

Complete Induction and the Well- Ordering Principle

Complete Induction and the Well- Ordering Principle Complete Induction and the Well- Ordering Principle Complete Induction as a Rule of Inference In mathematical proofs, complete induction (PCI) is a rule of inference of the form P (a) P (a + 1) P (b) k

More information

Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning. Mathematical Proof and Proving (MPP)

Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning. Mathematical Proof and Proving (MPP) Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning Terminology, Notations, Definitions, & Principles: Mathematical Proof and Proving (MPP) 1. A statement

More information

Theory of Computation

Theory of Computation Theory of Computation Prof. Michael Mascagni Florida State University Department of Computer Science 1 / 33 This course aims to cover... the development of computability theory using an extremely simple

More information

CS280, Spring 2004: Prelim Solutions

CS280, Spring 2004: Prelim Solutions CS280, Spring 2004: Prelim Solutions 1. [3 points] What is the transitive closure of the relation {(1, 2), (2, 3), (3, 1), (3, 4)}? Solution: It is {(1, 2), (2, 3), (3, 1), (3, 4), (1, 1), (2, 2), (3,

More information

Proof Techniques (Review of Math 271)

Proof Techniques (Review of Math 271) Chapter 2 Proof Techniques (Review of Math 271) 2.1 Overview This chapter reviews proof techniques that were probably introduced in Math 271 and that may also have been used in a different way in Phil

More information

DISCRETE MATH: FINAL REVIEW

DISCRETE 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 information

Supplementary Material for MTH 299 Online Edition

Supplementary 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 information

Chapter 1. Sets and Numbers

Chapter 1. Sets and Numbers Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write

More information

We want to show P (n) is true for all integers

We want to show P (n) is true for all integers Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to

More information

a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0. (x 1) 2 > 0.

a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0. (x 1) 2 > 0. For some problems, several sample proofs are given here. Problem 1. a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0.

More information

Math 210 Exam 2 - Practice Problems. 1. For each of the following, determine whether the statement is True or False.

Math 210 Exam 2 - Practice Problems. 1. For each of the following, determine whether the statement is True or False. Math 20 Exam 2 - Practice Problems. For each of the following, determine whether the statement is True or False. (a) {a,b,c,d} TRE (b) {a,b,c,d} FLSE (c) {a,b, } TRE (d) {a,b, } TRE (e) {a,b} {a,b} FLSE

More information

UNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS MATH 122: Logic and Foundations

UNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS MATH 122: Logic and Foundations UNIVERSITY OF VICTORIA DECEMBER EXAMINATIONS 2013 MATH 122: Logic and Foundations Instructor and section (check one): K. Mynhardt [A01] CRN 12132 G. MacGillivray [A02] CRN 12133 NAME: V00#: Duration: 3

More information

Solutions to Homework Set 1

Solutions to Homework Set 1 Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement

More information

CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:

CS2742 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 information

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3

More information

Discrete Structures CRN Test 3 Version 1 CMSC 2123 Autumn 2013

Discrete 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 2-2-203, in the space labeled DATE. 4. Print your CRN, 786, in the space

More information

can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4

can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4 .. Conditions for Injectivity and Surjectivity In this section, we discuss what we can say about linear maps T : R n R m given only m and n. We motivate this problem by looking at maps f : {,..., n} {,...,

More information

CSCE 222 Discrete Structures for Computing. Review for Exam 1. Dr. Hyunyoung Lee !!!

CSCE 222 Discrete Structures for Computing. Review for Exam 1. Dr. Hyunyoung Lee !!! CSCE 222 Discrete Structures for Computing Review for Exam 1 Dr. Hyunyoung Lee 1 Topics Propositional Logic (Sections 1.1, 1.2 and 1.3) Predicate Logic (Sections 1.4 and 1.5) Rules of Inferences and Proofs

More information

A Guide to Proof-Writing

A Guide to Proof-Writing A Guide to Proof-Writing 437 A Guide to Proof-Writing by Ron Morash, University of Michigan Dearborn Toward the end of Section 1.5, the text states that there is no algorithm for proving theorems.... Such

More information

Review for Midterm 1. Andreas Klappenecker

Review for Midterm 1. Andreas Klappenecker Review for Midterm 1 Andreas Klappenecker Topics Chapter 1: Propositional Logic, Predicate Logic, and Inferences Rules Chapter 2: Sets, Functions (Sequences), Sums Chapter 3: Asymptotic Notations and Complexity

More information

LECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel

LECTURE NOTES DISCRETE MATHEMATICS. Eusebius Doedel LECTURE NOTES on DISCRETE MATHEMATICS Eusebius Doedel 1 LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is

More information

1 of 8 7/15/2009 3:43 PM Virtual Laboratories > 1. Foundations > 1 2 3 4 5 6 7 8 9 6. Cardinality Definitions and Preliminary Examples Suppose that S is a non-empty collection of sets. We define a relation

More information

Mathematical Induction

Mathematical Induction Mathematical Induction MAT30 Discrete Mathematics Fall 018 MAT30 (Discrete Math) Mathematical Induction Fall 018 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

More information

Propositional Logic. What is discrete math? Tautology, equivalence, and inference. Applications

Propositional Logic. What is discrete math? Tautology, equivalence, and inference. Applications What is discrete math? Propositional Logic The real numbers are continuous in the senses that: between any two real numbers there is a real number The integers do not share this property. In this sense

More information

x P(x) x P(x) CSE 311: Foundations of Computing announcements last time: quantifiers, review: logical Inference Fall 2013 Lecture 7: Proofs

x P(x) x P(x) CSE 311: Foundations of Computing announcements last time: quantifiers, review: logical Inference Fall 2013 Lecture 7: Proofs CSE 311: Foundations of Computing Fall 2013 Lecture 7: Proofs announcements Reading assignment Logical inference 1.6-1.7 7 th Edition 1.5-1.7 6 th Edition Homework #2 due today last time: quantifiers,

More information

Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations

Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...

More information