Review 1. Andreas Klappenecker
|
|
- Jasper Lester Hicks
- 5 years ago
- Views:
Transcription
1 Review 1 Andreas Klappenecker
2 Summary Propositional Logic, Chapter 1 Predicate Logic, Chapter 1 Proofs, Chapter 1 Sets, Chapter 2 Functions, Chapter 2 Sequences and Sums, Chapter 2 Asymptotic Notations, Chapter 3 Time Complexity of Algorithms, Chapter 3
3 Strategy for Exam Preparation - Start studying now! - Make sure you know your definitions! - Do odd numbered exercises (Solutions are in the Appendix) - Review your quizzes - Review your homework
4 Logical Connectives - Summary Let B={t,f}. Assign to each connective a function M: B->B that determines its semantics. P M (P ) f t t f P Q M (P, Q) y if is false. M (P, Q) M (P, Q) M (P, Q) M (P, Q) f f f f f t t f t f t t t f t f f t t f f t t t t f t t
5 Alternative Summary Summary. Informally, we can summarize the meaning of the connectives as follows: 1) The and connective (a b) is true if and only if both a and b are true. 2) The or connective (a b) is true if and only if at least one of a, b is true. 3) The exclusive or (a b) is true if and only if precisely one of a, b is true. 4) The implication (a b) is false if and only if the premise a is true and the conclusion b is false. 5) The biconditional connective (a b) is true if and only if the truth values of a and b are the same. An interpretation of a subset S of Prop is an assignment of truth values to all variables that occur in the propositions contained in S. We showed that there exist a unique valuation extending an interpretation of all propositions.
6 Conditional Perhaps the most important logical connective is the conditional, also known as implication: p -> q The statement asserts that q holds on the condition that p holds. We call p the hypothesis or premise, and q the conclusion or consequence. Typical usage in proofs: If p, then q ; p implies q ; q only if p ; q when p ; q follows from p p is sufficient for q ; a sufficient condition for q is p ; a necessary condition for p is q ; q is necessary for p
7 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. E: Z->{t,f} with E(x)= x is an even integer, E(6) is true; O: Z->{t,f} with O(x)= x is an odd integer; O(6) is false.
8 Logical Equivalence Two statements involving quantifiers and predicates are logically equivalent if and only if they have the same truth values no matter which predicates are substituted into these statements and which domain is used. We write A B for logically equivalent A and B. You use logical equivalences to derive more convenient forms statements. Example: De Morgan s laws.
9 De Morgan s Laws xp (x) x P (x) xp (x) x P (x) (p q) p q (p q) p q
10 Valid Arguments An argument in propositional logic is a sequence of propositions that end with a proposition called conclusion. The argument is called valid if the conclusion follows from the preceding statements (called premises). In other words, in a valid argument it is impossible that all premises are true but the conclusion is false.
11 Modus Ponens The tautology (p (p->q)) -> q is the basis for the rule of inference called modus ponens. p p -> q q
12 Modus Tollens q p -> q p The University will not close on Wednesday. If it snows on Wednesday, then the University will close. Therefore, It will not snow on Wednesday
13 Simplification p q p
14 Formal Argument p q r p r s s t t Argument 1) p q Hypothesis 2) p Simplification of 1) 3) r p Hypothesis 4) r Modus tollens using 2) and 3) 5) r s Hyposthesis 6) s Modus ponens using 4), 5) 7) s t Hypothesis 8) t Modus ponens using 6), 7)
15 Sets, Functions, Summations
16 Set Builder Notation The set builder notation describes all elements as a subset of a set having a certain property. Q = { p/q R p Z, q Z, and q 0 } [a,b] = { x R a <= x <= b } [a,b) = { x R a <= x < b } (a,b] = { x R a < x <= b } (a,b) = { x R a < x < b } 16
17 Equality of Sets Two sets A and B are called equal if and only if they have the same elements. A = B if and only if x(x A x B) [To prove A=B, it is sufficient to show that both x(x A x B) and x(x B x A) hold. Why? ] 17
18 Subset A set A is a subset of B, written A B, if and only if every element of A is an element of B. Thus, A B if and only if x(x A x B) 18
19 Cardinality of a Set Let S be a set with a finite number of elements. We say that the set has cardinality n if and only if S contains n elements. We write S to denote the cardinality of the set. For example, = 0. 19
20 Power Sets Given a set S, the power set P(S) of S is the set of all subsets of S. Example: P( {1} ) = {, {1} } P( {1,2} ) = {, {1}, {2}, {1,2} } P( ) = { } since every set contains the empty set as a subset, even the empty set. P({ }) = {, { }}. 20
21 Cartesian Products Let A and B be sets. The Cartesian product of A and B, denote AxB, is the set of all pairs (a,b) with a A and b B. AxB = { (a,b) a A b B } 21
22 Set Operations Give two sets A and B. You should know - the union of A and B - the intersection of A and B - the set difference between A and B - the complement of A 22
23 De Morgan Laws A B = A B Proof : A B = {x x A B} by definition of complement = {x (x A B)} = {x (x A x B)} by definition of intersection = {x (x A) (x B)} de Morgan s law from logic = {x (x A) (x B)} by definition of = {x x A x B} by definition of complement = {x x A B} by definition of union = A B 23
24 Terminology Let f: A -> B be a function. We call - A the domain of f and - B the codomain of f. The range of f is the set f(a) = { f(a) a in A } 24
25 Functions Let A and B be sets. Consider a function f: A-> B. - When is f surjective? - When is f injective? - When is f bijective? 25
26 Floor Function The floor function : R -> Z assigns to a real number x the largest integer <= x. 3.2 = = = 3 26
27 Ceiling Function The ceiling function : R -> Z assigns to a real number x the smallest integer >= x. 3.2 = = = 1 27
28 Basic Facts We have x = n if and only if n <= x < n+1. We have x =n if and only if n-1< x <= n. We have x = n if and only if x-1 < n <= x. We have x =n if and only if x<= n < x+1. 28
29 Example Prove or disprove: x = x 29
30 Example 3 Let m = x Hence, m x <m+1 Thus, m 2 x < (m + 1) 2 It follows that m 2 x<(m + 1) 2 Therefore, m x<m+1 Thus, we can conclude that m = x This proves our claim. 30
31 Geometric Series Extremely useful! If a and r = 0 are real numbers, then n ar n+1 ar j a = r 1 if r = 1 (n + 1)a if r =1 j=0 Proof: The case r = 1 holds, since ar j = a for each of the n + 1 terms of the sum. The case r = 1 holds, since (r 1) n n j=0 arj = ar j+1 = j=0 n+1 ar j j=1 n ar j j=0 n ar j j=0 = ar n+1 a and dividing by (r 1) yields the claim.
32 Sum of First n Positive Integers. Extremely useful! For all n 1, we have n k = n(n + 1)/2 k=1 We prove this by induction. Basis step: For n = 1, we have 1 k=1 k = 1 = 1(1 + 1)/2.
33 Sum of the First n Positive Integers Induction Hypothesis: We assume that the claim holds for n 1. Induction Step: Assuming the Induction Hypothesis, we will show that the claim holds for n. n k=1 k = n + n 1 k=1 k = 2n/2 +(n 1)n/2 by Induction Hypothesis = 2n+n2 n 2 = n(n+1) 2 Therefore, the claim follows by induction on n.
34 Infinite Geometric Series Let x be a real number such that x < 1. Then k=0 x k = 1 1 x.
35 Infinite Geometric Series Since the sum of a geometric series satisfies we have k=0 n k=0 x k = lim n x k = xn+1 1 x 1, n k=0 As lim n x n+1 = 0, we get x k = lim n x n+1 1 x 1 x k = k=0 1 x 1 = 1 1 x.
36 Asymptotic Notations
37 Big Oh Notation Let f,g: N -> R be functions from the natural numbers to the set of real numbers. We write f O(g) if and only if there exists some real number n 0 and a positive real constant U such that f(n) <= U g(n) for all n satisfying n >= n 0
38 Big Ω We define f(n) = Ω(g(n)) if and only if there exists a constant L and a natural number n 0 such that L g(n) <= f(n) holds for all n >= n 0. In other words, f(n) = Ω(g(n)) if and only if g(n) = O(f(n)).
39 Big Θ We define f(n) = Θ(g(n)) if and only if there exist constants L and U and a natural number n 0 such that L g(n) <= f(n) <= U g(n) holds for all n >= n 0. In other words, f(n) = Θ(g(n)) if and only if f(n) = Ω(g(n)) and f(n) = O(g(n)).
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 informationReview 3. Andreas Klappenecker
Review 3 Andreas Klappenecker Final Exam Friday, May 4, 2012, starting at 12:30pm, usual classroom Topics Topic Reading Algorithms and their Complexity Chapter 3 Logic and Proofs Chapter 1 Logic and Proofs
More informationCSCE 222 Discrete Structures for Computing. Review for the Final. Hyunyoung Lee
CSCE 222 Discrete Structures for Computing Review for the Final! Hyunyoung Lee! 1 Final Exam Section 501 (regular class time 8:00am) Friday, May 8, starting at 1:00pm in our classroom!! Section 502 (regular
More informationReview 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 informationPredicate 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 informationCSCE 222 Discrete Structures for Computing
CSCE 222 Discrete Structures for Computing Sets and Functions Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Sets Sets are the most fundamental discrete structure on which all other discrete
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 information2.1 Sets. Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R.
2. Basic Structures 2.1 Sets Definition 1 A set is an unordered collection of objects. Important sets: N, Z, Z +, Q, R. Definition 2 Objects in a set are called elements or members of the set. A set is
More informationLECTURE 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 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 informationn Empty Set:, or { }, subset of all sets n Cardinality: V = {a, e, i, o, u}, so V = 5 n Subset: A B, all elements in A are in B
Discrete Math Review Discrete Math Review (Rosen, Chapter 1.1 1.7, 5.5) TOPICS Sets and Functions Propositional and Predicate Logic Logical Operators and Truth Tables Logical Equivalences and Inference
More informationLecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel
Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical
More informationFoundations 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 informationLECTURE 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 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 informationBackground 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 informationEquivalence 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 informationIntro to Logic and Proofs
Intro to Logic and Proofs Propositions A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. Examples: It is raining today. Washington
More informationFunctions. Given a function f: A B:
Functions Given a function f: A B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f.
More informationLogic, 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 informationSets McGraw-Hill Education
Sets A set is an unordered collection of objects. The objects in a set are called the elements, or members of the set. A set is said to contain its elements. The notation a A denotes that a is an element
More informationIntroduction Logic Inference. Discrete Mathematics Andrei Bulatov
Introduction Logic Inference Discrete Mathematics Andrei Bulatov Discrete Mathematics - Logic Inference 6-2 Previous Lecture Laws of logic Expressions for implication, biconditional, exclusive or Valid
More informationMAT115A-21 COMPLETE LECTURE NOTES
MAT115A-21 COMPLETE LECTURE NOTES NATHANIEL GALLUP 1. Introduction Number theory begins as the study of the natural numbers the integers N = {1, 2, 3,...}, Z = { 3, 2, 1, 0, 1, 2, 3,...}, and sometimes
More informationLogic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
More informationCSCE 222 Discrete Structures for Computing. Review for Exam 2. Dr. Hyunyoung Lee !!!
CSCE 222 Discrete Structures for Computing Review for Exam 2 Dr. Hyunyoung Lee 1 Strategy for Exam Preparation - Start studying now (unless have already started) - Study class notes (lecture slides and
More informationChapter Summary. Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.
Chapter 2 Chapter Summary Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (2.6) Section 2.1 Section Summary Definition of sets Describing
More informationWe 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 informationPacket #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 informationMathematics Review for Business PhD Students
Mathematics Review for Business PhD Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationMath 3336: Discrete Mathematics Practice Problems for Exam I
Math 3336: Discrete Mathematics Practice Problems for Exam I The upcoming exam on Tuesday, February 26, will cover the material in Chapter 1 and Chapter 2*. You will be provided with a sheet containing
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 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 informationFinal Exam Review. 2. Let A = {, { }}. What is the cardinality of A? Is
1. Describe the elements of the set (Z Q) R N. Is this set countable or uncountable? Solution: The set is equal to {(x, y) x Z, y N} = Z N. Since the Cartesian product of two denumerable sets is denumerable,
More informationCSCE 222 Discrete Structures for Computing. Propositional Logic. Dr. Hyunyoung Lee. !!!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Propositional Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Propositions A proposition is a declarative sentence that is either true or false
More information2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
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 informationLogic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More informationSteinhardt 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 informationChapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
More informationMATH 2001 MIDTERM EXAM 1 SOLUTION
MATH 2001 MIDTERM EXAM 1 SOLUTION FALL 2015 - MOON Do not abbreviate your answer. Write everything in full sentences. Except calculators, any electronic devices including laptops and cell phones are not
More informationCSE 20 DISCRETE MATH SPRING
CSE 20 DISCRETE MATH SPRING 2016 http://cseweb.ucsd.edu/classes/sp16/cse20-ac/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
More informationCSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition
More informationUndergraduate 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 information586 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 informationMath 13, Spring 2013, Lecture B: Midterm
Math 13, Spring 2013, Lecture B: Midterm Name Signature UCI ID # E-mail address Each numbered problem is worth 12 points, for a total of 84 points. Present your work, especially proofs, as clearly as possible.
More informationCOMP 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 informationPacket #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 informationTheorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)
Week 1: Logic Lecture 1, 8/1 (Sections 1.1 and 1.3) Examples of theorems and proofs Theorem (Pythagoras). Let ABC be a right triangle, with legs of lengths a and b, and hypotenuse of length c. Then a +
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationNotes 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 informationChapter 1, Logic and Proofs (3) 1.6. Rules of Inference
CSI 2350, Discrete Structures Chapter 1, Logic and Proofs (3) Young-Rae Cho Associate Professor Department of Computer Science Baylor University 1.6. Rules of Inference Basic Terminology Axiom: a statement
More informationMathematics Review for Business PhD Students Lecture Notes
Mathematics Review for Business PhD Students Lecture Notes Anthony M. Marino Department of Finance and Business Economics Marshall School of Business University of Southern California Los Angeles, CA 90089-0804
More informationAnnouncements. Exam 1 Review
Announcements Quiz today Exam Monday! You are allowed one 8.5 x 11 in cheat sheet of handwritten notes for the exam (front and back of 8.5 x 11 in paper) Handwritten means you must write them by hand,
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 informationDiscrete Mathematics. (c) Marcin Sydow. Sets. Set operations. Sets. Set identities Number sets. Pair. Power Set. Venn diagrams
Contents : basic definitions and notation A set is an unordered collection of its elements (or members). The set is fully specified by its elements. Usually capital letters are used to name sets and lowercase
More informationSets, Logic, Relations, and Functions
Sets, Logic, Relations, and Functions Andrew Kay September 28, 2014 Abstract This is an introductory text, not a comprehensive study; these notes contain mainly definitions, basic results, and examples.
More informationMath Fall 2014 Final Exam Solutions
Math 2001-003 Fall 2014 Final Exam Solutions Wednesday, December 17, 2014 Definition 1. The union of two sets X and Y is the set X Y consisting of all objects that are elements of X or of Y. The intersection
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 information4. Sets The language of sets. Describing a Set. c Oksana Shatalov, Fall Set-builder notation (a more precise way of describing a set)
c Oksana Shatalov, Fall 2018 1 4. Sets 4.1. The language of sets Set Terminology and Notation Set is a well-defined collection of objects. Elements are objects or members of the set. Describing a Set Roster
More informationChapter 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 informationCSCE 222 Discrete Structures for Computing. Dr. Hyunyoung Lee
CSCE 222 Discrete Structures for Computing Sequences and Summations Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Sequences 2 Sequences A sequence is a function from a subset of the set of
More informationChapter 2 - Basics Structures
Chapter 2 - Basics Structures 2.1 - Sets Definitions and Notation Definition 1 (Set). A set is an of. These are called the or of the set. We ll typically use uppercase letters to denote sets: S, A, B,...
More informationSets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions
Sets and Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Notation x A means that element x is a member of set A. x / A means that x is not a member of A.
More informationChapter Summary. Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability
Chapter 2 1 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability Sequences and Summations Types of Sequences Summation
More informationMath 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 informationA 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 information4. Sets The language of sets. Describing a Set. c Oksana Shatalov, Fall
c Oksana Shatalov, Fall 2017 1 4. Sets 4.1. The language of sets Set Terminology and Notation Set is a well-defined collection of objects. Elements are objects or members of the set. Describing a Set Roster
More informationSection 2.2 Set Operations. Propositional calculus and set theory are both instances of an algebraic system called a. Boolean Algebra.
Section 2.2 Set Operations Propositional calculus and set theory are both instances of an algebraic system called a Boolean Algebra. The operators in set theory are defined in terms of the corresponding
More informationMTH 299 In Class and Recitation Problems SUMMER 2016
MTH 299 In Class and Recitation Problems SUMMER 2016 Last updated on: May 13, 2016 MTH299 - Examples CONTENTS Contents 1 Week 1 3 1.1 In Class Problems.......................................... 3 1.2 Recitation
More informationDiscrete Mathematics
Discrete Mathematics Dr. Thomas Baird January 7, 2013 Contents 1 Logic 2 1.1 Statements.................................... 2 1.1.1 And, Or, Not.............................. 2 1.1.2 Implication...............................
More informationReview CHAPTER. 2.1 Definitions in Chapter Sample Exam Questions. 2.1 Set; Element; Member; Universal Set Partition. 2.
CHAPTER 2 Review 2.1 Definitions in Chapter 2 2.1 Set; Element; Member; Universal Set 2.2 Subset 2.3 Proper Subset 2.4 The Empty Set, 2.5 Set Equality 2.6 Cardinality; Infinite Set 2.7 Complement 2.8 Intersection
More informationDepartment 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 informationSETS AND FUNCTIONS JOSHUA BALLEW
SETS AND FUNCTIONS JOSHUA BALLEW 1. Sets As a review, we begin by considering a naive look at set theory. For our purposes, we define a set as a collection of objects. Except for certain sets like N, Z,
More informationCOMP1002 exam study sheet
COMP1002 exam study sheet Propositional statement: expression that has a truth value (true/false). It is a tautology if it is always true, contradiction if always false. Logic connectives: negation ( not
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 informationThe Logic of Compound Statements cont.
The Logic of Compound Statements cont. CSE 215, Computer Science 1, Fall 2011 Stony Brook University http://www.cs.stonybrook.edu/~cse215 Refresh from last time: Logical Equivalences Commutativity of :
More informationChapter 2 - Basics Structures MATH 213. Chapter 2: Basic Structures. Dr. Eric Bancroft. Fall Dr. Eric Bancroft MATH 213 Fall / 60
MATH 213 Chapter 2: Basic Structures Dr. Eric Bancroft Fall 2013 Dr. Eric Bancroft MATH 213 Fall 2013 1 / 60 Chapter 2 - Basics Structures 2.1 - Sets 2.2 - Set Operations 2.3 - Functions 2.4 - Sequences
More informationAutomata Theory and Formal Grammars: Lecture 1
Automata Theory and Formal Grammars: Lecture 1 Sets, Languages, Logic Automata Theory and Formal Grammars: Lecture 1 p.1/72 Sets, Languages, Logic Today Course Overview Administrivia Sets Theory (Review?)
More informationProofs. Joe Patten August 10, 2018
Proofs Joe Patten August 10, 2018 1 Statements and Open Sentences 1.1 Statements A statement is a declarative sentence or assertion that is either true or false. They are often labelled with a capital
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 informationTools for reasoning: Logic. Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications:
Tools for reasoning: Logic Ch. 1: Introduction to Propositional Logic Truth values, truth tables Boolean logic: Implications: 1 Why study propositional logic? A formal mathematical language for precise
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationsoftware design & management Gachon University Chulyun Kim
Gachon University Chulyun Kim 2 Outline Propositional Logic Propositional Equivalences Predicates and Quantifiers Nested Quantifiers Rules of Inference Introduction to Proofs 3 1.1 Propositional Logic
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
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 informationREVIEW FOR THIRD 3200 MIDTERM
REVIEW FOR THIRD 3200 MIDTERM PETE L. CLARK 1) Show that for all integers n 2 we have 1 3 +... + (n 1) 3 < 1 n < 1 3 +... + n 3. Solution: We go by induction on n. Base Case (n = 2): We have (2 1) 3 =
More informationFall 2014 CMSC250/250H Midterm II
Fall 2014 CMSC250/250H Midterm II Circle Your Section! 0101 (10am: 3120, Ladan) 0102 (11am: 3120, Ladan) 0103 (Noon: 3120, Peter) 0201 (2pm: 3120, Yi) 0202 (10am: 1121, Vikas) 0203 (11am: 1121, Vikas)
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 informationWith Question/Answer Animations. Chapter 2
With Question/Answer Animations Chapter 2 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Sequences and Summations Types of
More informationDiscrete 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 informationMathematical Reasoning Rules of Inference & Mathematical Induction. 1. Assign propositional variables to the component propositional argument.
Mathematical Reasoning Rules of Inference & Mathematical Induction Example. If I take the day off it either rains or snows 2. When It rains, my basement floods 3. When the basement floods or it snows,
More informationSets and Functions. (As we will see, in describing a set the order in which elements are listed is irrelevant).
Sets and Functions 1. The language of sets Informally, a set is any collection of objects. The objects may be mathematical objects such as numbers, functions and even sets, or letters or symbols of any
More informationCSC 125 :: Final Exam May 3 & 5, 2010
CSC 125 :: Final Exam May 3 & 5, 2010 Name KEY (1 5) Complete the truth tables below: p Q p q p q p q p q p q T T T T F T T T F F T T F F F T F T T T F F F F F F T T 6-15. Match the following logical equivalences
More informationToday s Topics. Methods of proof Relationships to logical equivalences. Important definitions Relationships to sets, relations Special functions
Today s Topics Set identities Methods of proof Relationships to logical equivalences Functions Important definitions Relationships to sets, relations Special functions Set identities help us manipulate
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More informationIntroduction to Decision Sciences Lecture 6
Introduction to Decision Sciences Lecture 6 Andrew Nobel September 21, 2017 Functions Functions Given: Sets A and B, possibly different Definition: A function f : A B is a rule that assigns every element
More informationSupplementary Logic Notes CSE 321 Winter 2009
1 Propositional Logic Supplementary Logic Notes CSE 321 Winter 2009 1.1 More efficient truth table methods The method of using truth tables to prove facts about propositional formulas can be a very tedious
More informationConjunction: 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 informationDiscrete Mathematics. Spring 2017
Discrete Mathematics Spring 2017 Previous Lecture Principle of Mathematical Induction Mathematical Induction: rule of inference Mathematical Induction: Conjecturing and Proving Climbing an Infinite Ladder
More informationWritten Homework #1: Analysis of Algorithms
Written Homework #1: Analysis of Algorithms CIS 121 Fall 2016 cis121-16fa-staff@googlegroups.com Due: Thursday, September 15th, 2015 before 10:30am (You must submit your homework online via Canvas. A paper
More information