3.4 Set Operations Given a set A, the complement (in the Universal set U) A c is the set of all elements of U that are not in A. So A c = {x x / A}.

Size: px
Start display at page:

Download "3.4 Set Operations Given a set A, the complement (in the Universal set U) A c is the set of all elements of U that are not in A. So A c = {x x / A}."

Transcription

1 3.4 Set Operations Given a set, the complement (in the niversal set ) c is the set of all elements of that are not in. So c = {x x /. (This type of picture is called a Venn diagram.) Example 39 Let = {1, 2, 3. If = N then c = {4, 5, 6,... If = Z then c = {..., 2, 1, 0, 4, 5,... So, again, the niversal set is important. Let and be sets. The union of and is the set of elements belonging to either or : = {x (x ) (x ). The intersection of and is the set of elements belonging to both and : = {x (x ) (x ). 1

2 The difference of and is the set of elements of that do not belong to : \ = {x (x ) (x / ) = {x (x ) (x c ) by definition of complement = c by definition of intersection. Note that in general it is not true that \ and \ are equal. The symmetric difference of and is the set of elements that belong to or but not both: = {x (x x ) (x / ( )) = ( ) \ ( ) = ( ) ( ) c by above result on difference. * Note that can easily be expressed in terms of the exclusive or,, defined in Question 5 Sheet 2. For 3.5 Set Laws = {x (x ) (x )) Recall = if, and only if, and. 2

3 To prove use proof by the pick-a-point method; i.e. take an arbitrary element u and assume u. Then use any known true statements, including properties of and to prove u. Example 40 Let = Z, and let P (x) be the predicate x 1 2 with R(x) the predicate x 2. Prove that {x R(x) {x P (x). Proof. Take any u {x R(x). This means that R(u) is TRE, i.e. R(u) is FLSE. Thus u < 2. ut u = Z. So u < 2 means that u = 1, 0 or 1. For these u we can calculate u 1, obtaining the values 2, 1 or 0. In all cases u 1 2 and so P (u) is true. Hence u {x P (x). This is true for all u {x R(x) hence {x R(x) {x P (x). Note: If we are given, we can try to show = by showing that the propositions u and u are equivalent, i.e. (u ) (u ) for all u. *To see why this suffices, assume we have managed to show that (u ) (u ) for all u. Then if u is true we must have that u is true, which is the definition of. Similarly, if u is true then u is also true and so. Hence =. s an example we can try to prove ( ) c = c c. Example se the oolean Laws for logic to prove ( ) c = c c. (If you are asked to use the laws for logic you cannot use the laws for sets!) Take any u. Then u ( ) c u / (y definition of c ) (u ) (y definition of / ) ((u ) (u )) (y definition of ) ( (u ) (u )) (De Morgan s law for logic) u / u / (y definition of / ) u c u c (y definition of c ) u c c (y definition of ) 3

4 Thus we see how one of De Morgan s law for logic gives one of the two De Morgan s Laws for sets: (a) ( ) c = c c, (b) ( ) c = c c. Similarly we can prove the distributive law ( C) = ( ) ( C), by making use of the distributive law for propositions. Example se the oolean Laws for logic to prove ( C) = ( ) ( C). Take any u. Then u ( C) (u ) (u C) by definition of (u ) ((u ) (u C)) by definition of ((u ) (u )) ((u ) (u C)) distributive law for propositions (u ) (u C) by definition of u ( ) ( C) by definition of. This is one of the results in The oolean Laws for Sets: ssume,, C, the universal set. a) = 1) b) = a) ( ) C = ( C) 2) b) ( ) C = ( C) a) ( C) = ( ) ( C) 3) b) ( C) = ( ) ( C) 4) a) = b) = 5) a) c = b) c =. Commutative laws ssociative laws Distributive laws We can use Venn diagrams to see that these results are reasonable though you cannot use Venn diagrams to prove these results. 4

5 Example 41 In (i) the set ( ) ( C) is represented by all the shaded region while in (ii) the set ( C) is represented by the darker shaded regions. So ( ) ( C) and ( C) have the same diagrams. Proofs can be given for all the laws though some are a little tricky. Example (i) Take any u. Then u (u ) (u ). ut is the empty set so u is false for all possible u, i.e. u is a contradiction or, in symbols, (u ) O. Thus u (u ) (u ) definition of (u ) O u law 4a for propositions. Hence we have used law 4a for logic to prove law 4a for sets. (ii) Note, since in any given problem all elements lie in the universal set, the proposition u is trivially true for all u, i.e. (u ) I. For an example take any u. Then u c (u ) (u c ) definition of (u ) (u / ) definition of complement (u ) ( (u )) definition of / I law 5a for propositions u I (u ) seen above. Thus c =. Hence we have used law 4a for logic to prove law 5a for sets. Hopefully you can now see why the laws for sets are identical to the laws for propositions. The laws can be used to simplify expressions: 5

6 Example 42 (i) (C c ) (C c c ) = C c (( ) ( c )) (law 3b), distributive = C c (( c ) ) (law 3b), distributive = C c ( ) (law 5a) = C c (law 4b) Compare with Ex 11. *(ii) y definition = ( ) \ ( ). From the Venn diagram it looks as if this should equal ( \ ) ( \ ). Can you use the laws to prove this? = ( ) \ ( ) (y definition of ) = ( ) ( ) c (y definition of \) = ( ) ( c c ) (y DeMorgan s law) = (( ) c ) (( ) c ) (Distributive law) = {( c ) ( c ) {( c ) ( c ) (Distributive law, again) = {φ ( c ) {( c ) φ (law 5b) = {( c ) {( c ) (law 4a) = {( \ ) {( \ ) (y definition of \) = ( \ ) ( \ ) (law 1b) Note how we have had to use the distributive laws to make the expressions more complicated before simplification. *dditional Material (Not covered in lectures) Just as for propositions it is possible using these laws (along with De Morgan s laws) to prove the following obvious results, seen earlier for propositions at the end of section ( c ) c =, = φ = φ. = = (Domination Laws) (Idempotent Laws) 6

4. Sets The language of sets. Describing a Set. c Oksana Shatalov, Fall Set-builder notation (a more precise way of describing a set)

4. 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 information

COMP Logic for Computer Scientists. Lecture 16

COMP Logic for Computer Scientists. Lecture 16 COMP 1002 Logic for Computer Scientists Lecture 16 5 2 J dmin stuff 2 due Feb 17 th. Midterm March 2 nd. Semester break next week! Puzzle: the barber In a certain village, there is a (male) barber who

More information

4. Sets The language of sets. Describing a Set. c Oksana Shatalov, Fall

4. 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 information

Show Your Work! Point values are in square brackets. There are 35 points possible. Some facts about sets are on the last page.

Show Your Work! Point values are in square brackets. There are 35 points possible. Some facts about sets are on the last page. Formal Methods Name: Key Midterm 2, Spring, 2007 Show Your Work! Point values are in square brackets. There are 35 points possible. Some facts about sets are on the last page.. Determine whether each of

More information

COMP Logic for Computer Scientists. Lecture 16

COMP Logic for Computer Scientists. Lecture 16 COMP 1002 Logic for Computer Scientists Lecture 16 5 2 J Proof by contradiction To prove x F(x), prove x F x FLSE niversal instantiation: let n be an arbitrary element of the domain S of x Suppose that

More information

Section 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. 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 information

2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Spring

2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Spring c Dr Oksana Shatalov, Spring 2015 1 2. Sets 2.1&2.2: Sets and Subsets. Combining Sets. Set Terminology and Notation DEFINITIONS: Set is well-defined collection of objects. Elements are objects or members

More information

CSE 20. Final Review. CSE 20: Final Review

CSE 20. Final Review. CSE 20: Final Review CSE 20 Final Review Final Review Representation of integers in base b Logic Proof systems: Direct Proof Proof by contradiction Contraposetive Sets Theory Functions Induction Final Review Representation

More information

With Question/Answer Animations. Chapter 2

With 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 information

18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015)

18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015) 18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015) 1. Introduction The goal for this course is to provide a quick, and hopefully somewhat gentle, introduction to the task of formulating

More information

Finite Math Section 6_1 Solutions and Hints

Finite Math Section 6_1 Solutions and Hints Finite Math Section 6_1 Solutions and Hints by Brent M. Dingle for the book: Finite Mathematics, 7 th Edition by S. T. Tan. DO NOT PRINT THIS OUT AND TURN IT IN!!!!!!!! This is designed to assist you in

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

Operations on Sets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, 2012

Operations on Sets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, 2012 on Sets Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, 2012 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) on Sets Gazihan Alankuş (Based on original slides

More information

CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics CS 2336 Discrete Mathematics Lecture 9 Sets, Functions, and Relations: Part I 1 What is a Set? Set Operations Identities Cardinality of a Set Outline Finite and Infinite Sets Countable and Uncountable

More information

CM10196 Topic 2: Sets, Predicates, Boolean algebras

CM10196 Topic 2: Sets, Predicates, Boolean algebras CM10196 Topic 2: Sets, Predicates, oolean algebras Guy McCusker 1W2.1 Sets Most of the things mathematicians talk about are built out of sets. The idea of a set is a simple one: a set is just a collection

More information

2.2: Logical Equivalence: The Laws of Logic

2.2: Logical Equivalence: The Laws of Logic Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q

More information

Lecture 3. Title goes here 1. level Networks. Boolean Algebra and Multi-level. level. level. level. level

Lecture 3. Title goes here 1. level Networks. Boolean Algebra and Multi-level. level. level. level. level Lecture 3 Dr Richard Reilly Dept. of Electronic & Electrical Engineering Room 53, Engineering uilding oolean lgebra and Multi- oolean algebra George oole, little formal education yet was a brilliant scholar.

More information

Discrete Mathematics. (c) Marcin Sydow. Sets. Set operations. Sets. Set identities Number sets. Pair. Power Set. Venn diagrams

Discrete 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 information

Proving simple set properties...

Proving simple set properties... Proving simple set properties... Part 1: Some examples of proofs over sets Fall 2013 Proving simple set properties... Fall 2013 1 / 17 Introduction Overview: Learning outcomes In this session we will...

More information

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA

CHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA CHPTER 3 LOGIC GTES & OOLEN LGER C H P T E R O U T C O M E S Upon completion of this chapter, student should be able to: 1. Describe the basic logic gates operation 2. Construct the truth table for basic

More information

A set is an unordered collection of objects.

A set is an unordered collection of objects. Section 2.1 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain

More information

SET THEORY. Disproving an Alleged Set Property. Disproving an Alleged Set. Example 1 Solution CHAPTER 6

SET THEORY. Disproving an Alleged Set Property. Disproving an Alleged Set. Example 1 Solution CHAPTER 6 CHAPTER 6 SET THEORY SECTION 6.3 Disproofs, Algebraic Proofs, and Boolean Algebras Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Disproving an Alleged

More information

CS 173: Discrete Structures. Eric Shaffer Office Hour: Wed. 1-2, 2215 SC

CS 173: Discrete Structures. Eric Shaffer Office Hour: Wed. 1-2, 2215 SC CS 173: Discrete Structures Eric Shaffer Office Hour: Wed. 1-2, 2215 SC shaffer1@illinois.edu Agenda Sets (sections 2.1, 2.2) 2 Set Theory Sets you should know: Notation you should know: 3 Set Theory -

More information

Venn Diagrams; Probability Laws. Notes. Set Operations and Relations. Venn Diagram 2.1. Venn Diagrams; Probability Laws. Notes

Venn Diagrams; Probability Laws. Notes. Set Operations and Relations. Venn Diagram 2.1. Venn Diagrams; Probability Laws. Notes Lecture 2 s; Text: A Course in Probability by Weiss 2.4 STAT 225 Introduction to Probability Models January 8, 2014 s; Whitney Huang Purdue University 2.1 Agenda s; 1 2 2.2 Intersection: the intersection

More information

Logic Synthesis and Verification

Logic Synthesis and Verification Logic Synthesis and Verification Boolean Algebra Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2014 1 2 Boolean Algebra Reading F. M. Brown. Boolean Reasoning:

More information

Set Theory Basics of Set Theory. 6.2 Properties of Sets and Element Argument. 6.3 Algebraic Proofs 6.4 Boolean Algebras.

Set Theory Basics of Set Theory. 6.2 Properties of Sets and Element Argument. 6.3 Algebraic Proofs 6.4 Boolean Algebras. 9/6/17 Mustafa Jarrar: Lecture Notes in Discrete Mathematics. Birzeit University Palestine 2015 Set Theory 6.1. Basics of Set Theory 6.2 Properties of Sets and Element Argument 6.3 Algebraic Proofs 6.4

More information

1.1 Introduction to Sets

1.1 Introduction to Sets Math 166 Lecture Notes - S. Nite 8/29/2012 Page 1 of 5 1.1 Introduction to Sets Set Terminology and Notation A set is a well-defined collection of objects. The objects are called the elements and are usually

More information

Theorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)

Theorem. 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 information

Introduction to Quantum Logic. Chris Heunen

Introduction to Quantum Logic. Chris Heunen Introduction to Quantum Logic Chris Heunen 1 / 28 Overview Boolean algebra Superposition Quantum logic Entanglement Quantum computation 2 / 28 Boolean algebra 3 / 28 Boolean algebra A Boolean algebra is

More information

CISC-102 Winter 2016 Lecture 17

CISC-102 Winter 2016 Lecture 17 CISC-102 Winter 2016 Lecture 17 Logic and Propositional Calculus Propositional logic was eventually refined using symbolic logic. The 17th/18th century philosopher Gottfried Leibniz (an inventor of calculus)

More information

2. Introduction to commutative rings (continued)

2. Introduction to commutative rings (continued) 2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of

More information

Axioms of Probability

Axioms of Probability Sample Space (denoted by S) The set of all possible outcomes of a random experiment is called the Sample Space of the experiment, and is denoted by S. Example 1.10 If the experiment consists of tossing

More information

Discrete Basic Structure: Sets

Discrete Basic Structure: Sets KS091201 MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) Discrete Basic Structure: Sets Discrete Math Team 2 -- KS091201 MD W-07 Outline What is a set? Set properties Specifying a set Often used sets The universal

More information

Solutions to Sample Problems for Midterm

Solutions to Sample Problems for Midterm Solutions to Sample Problems for Midterm Problem 1. The dual of a proposition is defined for which contains only,,. It is For a compound proposition that only uses,, as operators, we obtained the dual

More information

Equivalence and Implication

Equivalence and Implication Equivalence and Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February 7 8, 2018 Alice E. Fischer Laws of Logic... 1/33 1 Logical Equivalence Contradictions and Tautologies 2 3 4 Necessary

More information

Discrete Structures of Computer Science Propositional Logic III Rules of Inference

Discrete Structures of Computer Science Propositional Logic III Rules of Inference Discrete Structures of Computer Science Propositional Logic III Rules of Inference Gazihan Alankuş (Based on original slides by Brahim Hnich) July 30, 2012 1 Previous Lecture 2 Summary of Laws of Logic

More information

What is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box.

What is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box. Question 1. [10 marks] Part (a) [2 marks] What is the decimal (base 10) representation of the binary number 110101? Show your work and place your final answer in the box. 2 0 + 2 2 + 2 4 + 2 5 = 1 + 4

More information

Logic Gates and Boolean Algebra

Logic Gates and Boolean Algebra Logic Gates and oolean lgebra The ridge etween Symbolic Logic nd Electronic Digital Computing Compiled y: Muzammil hmad Khan mukhan@ssuet.edu.pk asic Logic Functions and or nand nor xor xnor not 2 Logic

More information

HW 4 SOLUTIONS. , x + x x 1 ) 2

HW 4 SOLUTIONS. , x + x x 1 ) 2 HW 4 SOLUTIONS The Way of Analysis p. 98: 1.) Suppose that A is open. Show that A minus a finite set is still open. This follows by induction as long as A minus one point x is still open. To see that A

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross

More information

Sets McGraw-Hill Education

Sets 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 information

A B is shaded A B A B

A B is shaded A B A B NION: Let and be subsets of a universal set. The union of sets and is the set of all elements in that belong to or to or to both, and is denoted. Symbolically: = {x x or x } EMMPLE: Let = {a, b, c, d,

More information

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.

Boolean 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 information

Set Operations. Combining sets into new sets

Set Operations. Combining sets into new sets Set Operations Combining sets into new sets Union of Sets The union of two sets is the set of all elements that are in one or the other set: A B = {x x A x B} The union is the set theoretic equivalent

More information

Homework 1 2/7/2018 SOLUTIONS Exercise 1. (a) Graph the following sets (i) C = {x R x in Z} Answer:

Homework 1 2/7/2018 SOLUTIONS Exercise 1. (a) Graph the following sets (i) C = {x R x in Z} Answer: Homework 1 2/7/2018 SOLTIONS Eercise 1. (a) Graph the following sets (i) C = { R in Z} nswer: 0 R (ii) D = {(, y), y in R,, y 2}. nswer: = 2 y y = 2 (iii) C C nswer: y 1 2 (iv) (C C) D nswer: = 2 y y =

More information

4 Quantifiers and Quantified Arguments 4.1 Quantifiers

4 Quantifiers and Quantified Arguments 4.1 Quantifiers 4 Quantifiers and Quantified Arguments 4.1 Quantifiers Recall from Chapter 3 the definition of a predicate as an assertion containing one or more variables such that, if the variables are replaced by objects

More information

CHAPTER 1 SETS AND EVENTS

CHAPTER 1 SETS AND EVENTS CHPTER 1 SETS ND EVENTS 1.1 Universal Set and Subsets DEFINITION: set is a well-defined collection of distinct elements in the universal set. This is denoted by capital latin letters, B, C, If an element

More information

CMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates

CMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates CMSC 33 Lecture 6 Postulates & Theorems of oolean lgebra Semiconductors CMOS Logic Gates UMC, CMSC33, Richard Chang Last Time Overview of second half of this course Logic gates & symbols

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

Review. Propositional Logic. Propositions atomic and compound. Operators: negation, and, or, xor, implies, biconditional.

Review. Propositional Logic. Propositions atomic and compound. Operators: negation, and, or, xor, implies, biconditional. Review Propositional Logic Propositions atomic and compound Operators: negation, and, or, xor, implies, biconditional Truth tables A closer look at implies Translating from/ to English Converse, inverse,

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

Example: Use a direct argument to show that the sum of two even integers has to be even. Solution: Recall that an integer is even if it is a multiple

Example: Use a direct argument to show that the sum of two even integers has to be even. Solution: Recall that an integer is even if it is a multiple Use a direct argument to show that the sum of two even integers has to be even. Solution: Recall that an integer is even if it is a multiple of 2, that is, an integer x is even if x = 2y for some integer

More information

Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra

Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra oolean lgebra Introduction 1854: Logical algebra was published by George oole known today as oolean lgebra It s a convenient way and systematic way of expressing and analyzing the operation of logic circuits.

More information

CSE 20 DISCRETE MATH. Winter

CSE 20 DISCRETE MATH. Winter CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Evaluate which proof technique(s) is appropriate for a given proposition Direct proof Proofs by contraposition

More information

BOOLEAN ALGEBRA INTRODUCTION SUBSETS

BOOLEAN ALGEBRA INTRODUCTION SUBSETS BOOLEAN ALGEBRA M. Ragheb 1/294/2018 INTRODUCTION Modern algebra is centered around the concept of an algebraic system: A, consisting of a set of elements: ai, i=1, 2,, which are combined by a set of operations

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

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

Sets. Introduction to Set Theory ( 2.1) Basic notations for sets. Basic properties of sets CMSC 302. Vojislav Kecman

Sets. Introduction to Set Theory ( 2.1) Basic notations for sets. Basic properties of sets CMSC 302. Vojislav Kecman Introduction to Set Theory ( 2.1) VCU, Department of Computer Science CMSC 302 Sets Vojislav Kecman A set is a new type of structure, representing an unordered collection (group, plurality) of zero or

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

1.1 Language and Logic

1.1 Language and Logic c Oksana Shatalov, Spring 2018 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,

More information

CSC Discrete Math I, Spring Propositional Logic

CSC Discrete Math I, Spring Propositional Logic CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)

More information

Sets, Proofs and Functions. Anton Velinov

Sets, Proofs and Functions. Anton Velinov Sets, Proofs and Functions Anton Velinov DIW WS 2018 Takeaways Sets and set operations. De Morgan s laws. Logical quantifiers and operators. Implication. Methods of proof, deduction, induction, contradiction.

More information

Exclusive Disjunction

Exclusive Disjunction Exclusive Disjunction Recall A statement is a declarative sentence that is either true or false, but not both. If we have a declarative sentence s, p: s is true, and q: s is false, can we rewrite s is

More information

Today s topics. Introduction to Set Theory ( 1.6) Naïve set theory. Basic notations for sets

Today s topics. Introduction to Set Theory ( 1.6) Naïve set theory. Basic notations for sets Today s topics Introduction to Set Theory ( 1.6) Sets Definitions Operations Proving Set Identities Reading: Sections 1.6-1.7 Upcoming Functions A set is a new type of structure, representing an unordered

More information

Definition 2. Conjunction of p and q

Definition 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 information

Section 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.

Section 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false. Section 1.2 Propositional Equivalences A tautology is a proposition which is always true. Classic Example: P P A contradiction is a proposition which is always false. Classic Example: P P A contingency

More information

211 Real Analysis. f (x) = x2 1. x 1. x 2 1

211 Real Analysis. f (x) = x2 1. x 1. x 2 1 Part. Limits of functions. Introduction 2 Real Analysis Eample. What happens to f : R \ {} R, given by f () = 2,, as gets close to? If we substitute = we get f () = 0 which is undefined. Instead we 0 might

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

Solutions to Homework I (1.1)

Solutions to Homework I (1.1) Solutions to Homework I (1.1) Problem 1 Determine whether each of these compound propositions is satisable. a) (p q) ( p q) ( p q) b) (p q) (p q) ( p q) ( p q) c) (p q) ( p q) (a) p q p q p q p q p q (p

More information

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

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

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

CSC165 Mathematical Expression and Reasoning for Computer Science

CSC165 Mathematical Expression and Reasoning for Computer Science CSC165 Mathematical Expression and Reasoning for Computer Science Lisa Yan Department of Computer Science University of Toronto January 21, 2015 Lisa Yan (University of Toronto) Mathematical Expression

More information

CSE 20 DISCRETE MATH WINTER

CSE 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 information

CSE 20 DISCRETE MATH SPRING

CSE 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 information

Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur. Lecture 1 Real Numbers

Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur. Lecture 1 Real Numbers Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Lecture 1 Real Numbers In these lectures, we are going to study a branch of mathematics called

More information

Chapter Summary. Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability

Chapter 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 information

PHIL12A Section answers, 16 February 2011

PHIL12A Section answers, 16 February 2011 PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B (A B) A A B T T T T T T T T T T T F

More information

1.1 Statements and Compound Statements

1.1 Statements and Compound Statements Chapter 1 Propositional Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something

More information

Math 730 Homework 6. Austin Mohr. October 14, 2009

Math 730 Homework 6. Austin Mohr. October 14, 2009 Math 730 Homework 6 Austin Mohr October 14, 2009 1 Problem 3A2 Proposition 1.1. If A X, then the family τ of all subsets of X which contain A, together with the empty set φ, is a topology on X. Proof.

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

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.

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. 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 information

CHAPTER 1. Preliminaries. 1 Set Theory

CHAPTER 1. Preliminaries. 1 Set Theory CHAPTER 1 Preliminaries 1 et Theory We assume that the reader is familiar with basic set theory. In this paragraph, we want to recall the relevant definitions and fix the notation. Our approach to set

More information

Lecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)

Lecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook) Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or

More information

Introduction to Set Operations

Introduction to Set Operations Introduction to Set Operations CIS008-2 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 12:00, Friday 21 st October 2011 Outline 1 Recap 2 Introduction to sets 3 Class Exercises

More information

Introduction to Sets and Logic (MATH 1190)

Introduction to Sets and Logic (MATH 1190) Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition

More information

Boolean Algebra. Boolean Variables, Functions. NOT operation. AND operation. AND operation (cont). OR operation

Boolean Algebra. Boolean Variables, Functions. NOT operation. AND operation. AND operation (cont). OR operation oolean lgebra asic mathematics for the study of logic design is oolean lgebra asic laws of oolean lgebra will be implemented as switching devices called logic gates. Networks of Logic gates allow us to

More information

1.3 Propositional Equivalences

1.3 Propositional Equivalences 1 1.3 Propositional Equivalences The replacement of a statement with another statement with the same truth is an important step often used in Mathematical arguments. Due to this methods that produce propositions

More information

Propositional logic (revision) & semantic entailment. p. 1/34

Propositional logic (revision) & semantic entailment. p. 1/34 Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)

More information

Set Theory. CSE 215, Foundations of Computer Science Stony Brook University

Set 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 information

UNIT 8A Computer Circuitry: Layers of Abstraction. Boolean Logic & Truth Tables

UNIT 8A Computer Circuitry: Layers of Abstraction. Boolean Logic & Truth Tables UNIT 8 Computer Circuitry: Layers of bstraction 1 oolean Logic & Truth Tables Computer circuitry works based on oolean logic: operations on true (1) and false (0) values. ( ND ) (Ruby: && ) 0 0 0 0 0 1

More information

3.1 Universal quantification and implication again. Claim 1: If an employee is male, then he makes less than 55,000.

3.1 Universal quantification and implication again. Claim 1: If an employee is male, then he makes less than 55,000. Chapter 3 Logical Connectives 3.1 Universal quantification and implication again So far we have considered an implication to be universal quantication in disguise: Claim 1: If an employee is male, then

More information

Announcement. Homework 1

Announcement. Homework 1 Announcement I made a few small changes to the course calendar No class on Wed eb 27 th, watch the video lecture Quiz 8 will take place on Monday April 15 th We will submit assignments using Gradescope

More information

Mathematics Review for Business PhD Students

Mathematics 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 information

MACM 101 Discrete Mathematics I. Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class)

MACM 101 Discrete Mathematics I. Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class) MACM 101 Discrete Mathematics I Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class) Reminder: the work you submit must be your own. Any collaboration and

More information

Notes on Sets for Math 10850, fall 2017

Notes on Sets for Math 10850, fall 2017 Notes on Sets for Math 10850, fall 2017 David Galvin, University of Notre Dame September 14, 2017 Somewhat informal definition Formally defining what a set is is one of the concerns of logic, and goes

More information

Math Final Exam December 14, 2009 Page 1 of 5

Math Final Exam December 14, 2009 Page 1 of 5 Math 201-803-Final Exam December 14, 2009 Page 1 of 5 (3) 1. Evaluate the expressions: (a) 10 C 4 (b) 10 P 4 (c) 15!4! 3!11! (4) 2. (a) In how many ways can a president, a vice president and a treasurer

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

CISC-102 Fall 2018 Week 11

CISC-102 Fall 2018 Week 11 page! 1 of! 26 CISC-102 Fall 2018 Pascal s Triangle ( ) ( ) An easy ( ) ( way ) to calculate ( ) a table of binomial coefficients was recognized centuries ago by mathematicians in India, ) ( ) China, Iran

More information

A generalization of modal definability

A generalization of modal definability A generalization of modal definability Tin Perkov Polytechnic of Zagreb Abstract. Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models

More information