Discrete Mathematical Structures: Theory and Applications

Size: px
Start display at page:

Download "Discrete Mathematical Structures: Theory and Applications"

Transcription

1 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 Venn diagrams Learn how to represent sets in computer memory Learn about statements (propositions) Discrete Mathematical Structures: Theory and Applications 2

2 Learning Objectives Learn how to use logical connectives to combine statements Explore how to draw conclusions using various argument forms Become familiar with quantifiers and predicates Learn various proof techniques Explore what an algorithm is Discrete Mathematical Structures: Theory and Applications 3 Sets Definition: Well-defined collection of distinct objects Members or Elements: part of the collection Roster Method: Description of a set by listing the elements, enclosed with braces Examples: Vowels = {a,e,i,o,u} Primary colors = {red, blue, yellow} Membership examples a belongs to the set of Vowels is written as: a Vowels j does not belong to the set of Vowels: j Vowels Discrete Mathematical Structures: Theory and Applications 4

3 Sets Set-builder method A = { x x S, P(x) } or A = { x S P(x) } A is the set of all elements x of S, such that x satisfies the property P Example: If X = {2,4,6,8,10}, then in set-builder notation, X can be described as X = {n Z n is even and 2 n 10} Discrete Mathematical Structures: Theory and Applications 5 Sets Standard Symbols which denote sets of numbers N : The set of all natural numbers (i.e.,all positive integers) Z : The set of all integers Z* : The set of all nonzero integers E : The set of all even integers Q : The set of all rational numbers Q* : The set of all nonzero rational numbers Q + : The set of all positive rational numbers R : The set of all real numbers R* : The set of all nonzero real numbers R + : The set of all positive real numbers C : The set of all complex numbers C* : The set of all nonzero complex numbers Discrete Mathematical Structures: Theory and Applications 6

4 Sets Subsets X is a subset of Y is written as X Y X is not a subset of Y is written as X Y Example: X = {a,e,i,o,u}, Y = {a, i, u} and z = {b,c,d,f,g} Y X, since every element of Y is an element of X Y Z, since a Y, but a Z Discrete Mathematical Structures: Theory and Applications 7 Superset Sets X and Y are sets. If X Y, then X is contained in Y or Y contains X or Y is a superset of X, written Y X Proper Subset X and Y are sets. X is a proper subset of Y if X Y and there exists at least one element in Y that is not in X. This is written X Y. Example: X = {a,e,i,o,u}, Y = {a,e,i,o,u,y} X Y, since y Y, but y X Discrete Mathematical Structures: Theory and Applications 8

5 Set Equality Sets X and Y are sets. They are said to be equal if every element of X is an element of Y and every element of Y is an element of X, i.e. X Y and Y X Examples: {1,2,3} = {2,3,1} X = {red, blue, yellow} and Y = {c c is a primary color} Therefore, X=Y Empty (Null) Set A Set is Empty (Null) if it contains no elements. The Empty Set is written as The Empty Set is a subset of every set Discrete Mathematical Structures: Theory and Applications 9 Sets Finite and Infinite Sets X is a set. If there exists a nonnegative integer n such that X has n elements, then X is called a finite set with n elements. If a set is not finite, then it is an infinite set. Examples: Y = {1,2,3} is a finite set P = {red, blue, yellow} is a finite set E, the set of all even integers, is an infinite set, the Empty Set, is a finite set with 0 elements Discrete Mathematical Structures: Theory and Applications 10

6 Cardinality of Sets Sets Let S be a finite set with n distinct elements, where n 0. Then S = n, where the cardinality (number of elements) of S is n Example: If P = {red, blue, yellow}, then P = 3 Singleton A set with only one element is a singleton Example: H = { 4 }, H = 1, H is a singleton Discrete Mathematical Structures: Theory and Applications 11 Power Set Sets For any set X,the power set of X,written P(X),is the set of all subsets of X Example: If X = {red, blue, yellow}, then P(X) = {, {red}, {blue}, {yellow}, {red,blue}, {red, yellow}, {blue, yellow}, {red, blue, yellow} } Universal Set An arbitrarily chosen, but fixed set Discrete Mathematical Structures: Theory and Applications 12

7 Sets Venn Diagrams Abstract visualization of a Universal set, U as a rectangle, with all subsets of U shown as circles. Shaded portion represents the corresponding set Example: In Figure 1, Set X, shaded, is a subset of the Universal set, U Discrete Mathematical Structures: Theory and Applications 13 Union of Sets Sets Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X Y = {1,2,3,4,5,6,7,8,9} Discrete Mathematical Structures: Theory and Applications 14

8 Sets Intersection of Sets Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X Y = {5} Discrete Mathematical Structures: Theory and Applications 15 Disjoint Sets Sets Example: If X = {1,2,3,4,} and Y = {6,7,8,9}, then X Y = Discrete Mathematical Structures: Theory and Applications 16

9 Sets Discrete Mathematical Structures: Theory and Applications 17 Sets Discrete Mathematical Structures: Theory and Applications 18

10 Sets The union and intersection of three,four,or even infinitely many sets can be considered For a finite collection of n sets, X 1, X 2, X n where n 2 : Discrete Mathematical Structures: Theory and Applications 19 Index Set Sets Discrete Mathematical Structures: Theory and Applications 20

11 Sets Example: If A = {a,b,c}, B = {x, y, z} and C = {1,2,3} then A B = and B C = and A C =. Therefore, A,B,C are pairwise disjoint Discrete Mathematical Structures: Theory and Applications 21 Difference Sets Example: If X = {a,b,c,d} and Y = {c,d,e,f}, then X Y = {a,b} and Y X = {e,f} Discrete Mathematical Structures: Theory and Applications 22

12 Complement Sets Example: If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then X = {a,b} Discrete Mathematical Structures: Theory and Applications 23 Sets Discrete Mathematical Structures: Theory and Applications 24

13 Sets Discrete Mathematical Structures: Theory and Applications 25 Sets Discrete Mathematical Structures: Theory and Applications 26

14 Sets Ordered Pair X and Y are sets. If x X and y Y, then an ordered pair is written (x,y) Order of elements is important. (x,y) is not necessarily equal to (y,x) Cartesian Product The Cartesian product of two sets X and Y,written X Y,is the set X Y ={(x,y) x X, y Y} For any set X, X = = X Example: X = {a,b}, Y = {c,d} X Y = {(a,c), (a,d), (b,c), (b,d)} Y X = {(c,a), (d,a), (c,b), (d,b)} Discrete Mathematical Structures: Theory and Applications 27 Diagonal of a Set Sets For a set X,the set δ x, is the diagonal of X, defined by δ x = {(x,x) x X} Example: X = {a,b,c}, δ x = {(a,a), (b,b), (c,c)} Discrete Mathematical Structures: Theory and Applications 28

15 Sets Computer Representation of Sets A Set may be stored in a computer in an array as an unordered list Problem: Difficult to perform operations on the set. Solution: use Bit Strings A Bit String is a sequence of 0s and 1s Length of a Bit String is the number of digits in the string Elements appear in order in the bit string A 0 indicates an element is absent, a 1 indicates that the element is present Discrete Mathematical Structures: Theory and Applications 29 Mathematical Logic Definition: Methods of reasoning, provides rules and techniques to determine whether an argument is valid Theorem: a statement that can be shown to be true (under certain conditions) Example: If x is an even integer, then x + 1 is an odd integer This statement is true under the condition that x is an integer is true Discrete Mathematical Structures: Theory and Applications 30

16 Mathematical Logic A statement, or a proposition, is a declarative sentence that is either true or false, but not both Lowercase letters denote propositions Examples: p: 2 is an even number (true) q: 3 is an odd number (true) r: A is a consonant (false) The following are not propositions: p: My cat is beautiful q: Are you in charge? Discrete Mathematical Structures: Theory and Applications 31 Mathematical Logic Truth value One of the values truth or falsity assigned to a statement True is abbreviated to T or 1 False is abbreviated to F or 0 Negation The negation of p, written p, is the statement obtained by negating statement p Truth values of p and p are opposite Symbol ~ is called not ~p is read as as not p Example: p: A is a consonant ~p: it is the case that A is not a consonant q: Are you in charge? Discrete Mathematical Structures: Theory and Applications 32

17 Mathematical Logic Truth Table Conjunction Let p and q be statements.the conjunction of p and q, written p ^ q, is the statement formed by joining statements p and q using the word and The statement p q is true if both p and q are true; otherwise p q is false Discrete Mathematical Structures: Theory and Applications 33 Mathematical Logic Conjunction Truth Table for Conjunction: Discrete Mathematical Structures: Theory and Applications 34

18 Mathematical Logic Disjunction Let p and q be statements. The disjunction of p and q, written p q, is the statement formed by joining statements p and q using the word or The statement p q is true if at least one of the statements p and q is true; otherwise p q is false The symbol is read or Discrete Mathematical Structures: Theory and Applications 35 Mathematical Logic Disjunction Truth Table for Disjunction: Discrete Mathematical Structures: Theory and Applications 36

19 Mathematical Logic Implication Let p and q be statements.the statement if p then q is called an implication or condition. The implication if p then q is written p q p q is read: If p, then q p is sufficient for q q if p q whenever p Discrete Mathematical Structures: Theory and Applications 37 Implication Mathematical Logic Truth Table for Implication: p is called the hypothesis, q is called the conclusion Discrete Mathematical Structures: Theory and Applications 38

20 Implication Mathematical Logic Let p: Today is Sunday and q: I will wash the car. The conjunction p q is the statement: p q : If today is Sunday, then I will wash the car The converse of this implication is written q p If I wash the car, then today is Sunday The inverse of this implication is ~p ~q If today is not Sunday, then I will not wash the car The contrapositive of this implication is ~q ~p If I do not wash the car, then today is not Sunday Discrete Mathematical Structures: Theory and Applications 39 Mathematical Logic Biimplication Let p and q be statements. The statement p if and only if q is called the biimplication or biconditional of p and q The biconditional p if and only if q is written p q p q is read: p if and only if q p is necessary and sufficient for q q if and only if p q when and only when p Discrete Mathematical Structures: Theory and Applications 40

21 Biconditional Mathematical Logic Truth Table for the Biconditional: Discrete Mathematical Structures: Theory and Applications 41 Mathematical Logic Statement Formulas Definitions Symbols p,q,r,...,called statement variables Symbols ~,,,,and are called logical connectives 1) A statement variable is a statement formula 2) If A and B are statement formulas, then the expressions (~A ), (A B), (A B ), (A B ) and (A B ) are statement formulas Expressions are statement formulas that are constructed only by using 1) and 2) above Discrete Mathematical Structures: Theory and Applications 42

22 Mathematical Logic Precedence of logical connectives is: ~ highest second highest third highest fourth highest fifth highest Discrete Mathematical Structures: Theory and Applications 43 Example: Mathematical Logic Let A be the statement formula (~(p q )) (q p ) Truth Table for A is: Discrete Mathematical Structures: Theory and Applications 44

23 Tautology Mathematical Logic A statement formula A is said to be a tautology if the truth value of A is T for any assignment of the truth values T and F to the statement variables occurring in A Contradiction A statement formula A is said to be a contradiction if the truth value of A is F for any assignment of the truth values T and F to the statement variables occurring in A Discrete Mathematical Structures: Theory and Applications 45 Logically Implies Mathematical Logic A statement formula A is said to logically imply a statement formula B if the statement formula A B is a tautology. If A logically implies B, then symbolically we write A B Logically Equivalent A statement formula A is said to be logically equivalent to a statement formula B if the statement formula A B is a tautology. If A is logically equivalent to B, then symbolically we write A B (or A B) Discrete Mathematical Structures: Theory and Applications 46

24 Mathematical Logic Discrete Mathematical Structures: Theory and Applications 47 Mathematical Logic Proof of (~p q ) (~(q p )) Discrete Mathematical Structures: Theory and Applications 48

25 Mathematical Logic Proof of (~p q ) (~(q p )) [Continued] Discrete Mathematical Structures: Theory and Applications 49 Validity of Arguments Proof: an argument or a proof of a theorem consists of a finite sequence of statements ending in a conclusion A, A, A,..., A, n A 1 n Argument: a finite sequence of statements. The final statement, An, is the conclusion, and the statements A, A, A,..., An 1 are the premises of the argument. An argument is logically valid if the statement formula,,,..., n 1 n A A A A is a tautology. Discrete Mathematical Structures: Theory and Applications 50 A

26 Validity of Arguments Valid Argument Forms Modus Ponens (Method of Affirming) Modus Tollens (Method of Denying) Discrete Mathematical Structures: Theory and Applications 51 Validity of Arguments Valid Argument Forms Disjunctive Syllogisms Disjunctive Syllogisms Discrete Mathematical Structures: Theory and Applications 52

27 Valid Argument Forms Hypothetical Syllogism Validity of Arguments Dilemma Discrete Mathematical Structures: Theory and Applications 53 Validity of Arguments Valid Argument Forms Conjunctive Simplification Conjunctive Simplification Discrete Mathematical Structures: Theory and Applications 54

28 Valid Argument Forms Disjunctive Addition Validity of Arguments Disjunctive Addition Discrete Mathematical Structures: Theory and Applications 55 Validity of Arguments Valid Argument Forms Conjunctive Addition Discrete Mathematical Structures: Theory and Applications 56

29 Quantifiers and First Order Logic Predicate or Propositional Function Let x be a variable and D be a set; P(x) is a sentence Then P(x) is called a predicate or propositional function with respect to the set D if for each value of x in D, P(x) is a statement; i.e., P(x) is true or false Moreover, D is called the domain of the discourse and x is called the free variable Discrete Mathematical Structures: Theory and Applications 57 Quantifiers and First Order Logic Predicate or Propositional Function Example: Q(x,y) : x > y, where the Domain is the set of integers Q is a 2-place predicate Q is T for Q(4,3) and Q is F for Q (3,4) Discrete Mathematical Structures: Theory and Applications 58

30 Quantifiers and First Order Logic Universal Quantifier Let P(x) be a predicate and let D be the domain of the discourse. The universal quantification of P(x) is the statement: For all x, P(x) For every x, P(x) The symbol x P(x) or Two-place predicate: is read as for all and every x y P( x, y) Discrete Mathematical Structures: Theory and Applications 59 Quantifiers and First Order Logic Existential Quantifier Let P(x) be a predicate and let D be the domain of the discourse. The existential quantification of P(x) is the statement: There exists x, P(x) The symbol x P(x) Bound Variable The variable appearing in: is read as there exists x P(x) or x P(x) Discrete Mathematical Structures: Theory and Applications 60

31 Quantifiers and First Order Logic Negation of Predicates (DeMorgan s Laws) ~ x P( x) x ~ P( x) Example: If P(x) is the statement x has won a race where the domain of discourse is all runners, then the universal quantification of P(x) is x P(x), i.e., every runner has won a race. The negation of this statement is it is not the case that every runner has won a race. Therefore there exists at least one runner who has not won a race. Therefore: x P and so, ~ x P( x) x ~ P( x) ~ ( x) Discrete Mathematical Structures: Theory and Applications 61 Quantifiers and First Order Logic Negation of Predicates (DeMorgan s Laws) ~ x P( x) x ~ P( x) Discrete Mathematical Structures: Theory and Applications 62

32 Quantifiers and First Order Logic Formulas in Predicate Logic All statement formulas are considered formulas Each n, n =1,2,...,n-place predicate P( x, x,..., 1 2 xn) containing the variables x, x,..., 1 2 xn is a formula. If A and B are formulas, then the expressions ~A, (A B), (A B), A B and A B are statement formulas, where ~,,, and are logical connectives If A is a formula and x is a variable, then x A(x) and x A(x) are formulas All formulas constructed using only above rules are considered formulas in predicate logic Discrete Mathematical Structures: Theory and Applications 63 Quantifiers and First Order Logic Additional Rules of Inference If the statement x P(x) is assumed to be true, then P(a) is also true,where a is an arbitrary member of the domain of the discourse. This rule is called the universal specification (US) If P(a) is true, where a is an arbitrary member of the domain of the discourse, then x P(x) is true. This rule is called the universal generalization (UG) If the statement x P (x) is true, then P(a) is true, for some member of the domain of the discourse. This rule is called the existential specification (ES) If P(a) is true for some member a of the domain of the discourse, then x P(x) is also true. This rule is called the existential generalization (EG) Discrete Mathematical Structures: Theory and Applications 64

33 Quantifiers and First Order Logic Counterexample An argument has the form x (P(x ) Q(x )), where the domain of discourse is D To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of the above implication To show that x (P(x) Q(x)) is false by finding an x in D such that P(x) Q(x) is false is called the disproof of the given statement by counterexample Discrete Mathematical Structures: Theory and Applications 65 Theorem Proof Techniques Statement that can be shown to be true (under certain conditions) Typically Stated in one of three ways As Facts As Implications As Biimplications Discrete Mathematical Structures: Theory and Applications 66

34 Proof Techniques Direct Proof or Proof by Direct Method Proof of those theorems that can be expressed in the form x (P(x) Q(x)), D is the domain of discourse Select a particular, but arbitrarily chosen, member a of the domain D Show that the statement P(a) Q(a) is true. (Assume that P(a) is true Show that Q(a) is true By the rule of Universal Generalization (UG), x (P(x) Q(x)) is true Discrete Mathematical Structures: Theory and Applications 67 Indirect Proof Proof Techniques The implication p q is equivalent to the implication ( q p) Therefore, in order to show that p q is true, one can also show that the implication ( q p) is true To show that ( q p) is true, assume that the negation of q is true and prove that the negation of p is true Discrete Mathematical Structures: Theory and Applications 68

35 Proof Techniques Proof by Contradiction Assume that the conclusion is not true and then arrive at a contradiction Example: Prove that there are infinitely many prime numbers Proof: Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p 1,p 2,,p n Consider the number q = p 1 p 2 p n +1. q is not divisible by any of the listed primes Therefore, q is a prime. However, it was not listed. Contradiction! Therefore, there are infinitely many primes Discrete Mathematical Structures: Theory and Applications 69 Proof Techniques Discrete Mathematical Structures: Theory and Applications 70

36 Proof Techniques Proof of Biimplications To prove a theorem of the form x (P(x) Q(x )), where D is the domain of the discourse, consider an arbitrary but fixed element a from D. For this a, prove that the biimplication P(a) Q(a) is true The biimplication p q is equivalent to (p q) (q p) Prove that the implications p q and q p are true Assume that p is true and show that q is true Assume that q is true and show that p is true Discrete Mathematical Structures: Theory and Applications 71 Proof Techniques Proof of Equivalent Statements Consider the theorem that says that statements p,q and r are equivalent Show that p q, q r and r p Assume p and prove q. Then assume q and prove r Finally, assume r and prove p Or, prove that p if and only if q, and then q if and only if r Other methods are possible Discrete Mathematical Structures: Theory and Applications 72

37 Algorithms Definition: step-by-step problem-solving process in which a solution is arrived at in a finite amount of time All algorithms have the following properties: Input : For example, a set of numbers to find the sum of the numbers Output : For example, the sum of the numbers Precision : Each step of the algorithm is precisely defined Uniqueness : Results of each step are unique and depend on the input and results of previous step Finiteness : Algorithm must terminate after executing a finite number of steps Generality : Algorithm is general in that it applies to a set of inputs Discrete Mathematical Structures: Theory and Applications 73 Algorithms Pseudocode Conventions The symbol := is called the assignment operator Example: The statement x := a is read as assign the value a to x or x gets the value a or copy the value of a into x x := a is also known as an assignment statement Control Structures One way-selection if booleanexpression then statement If booleanexpression evaluates to true, statement is evaluated Two way-selection if booleanexpression then statement1 else statement2 If booleanexpression evaluates to true, statement1 executes, otherwise statement2 executes Discrete Mathematical Structures: Theory and Applications 74

38 Algorithms Pseudocode Conventions Control Structures The while loop takes the form: while booleanexpression do loopbody The booleanexpression is evaluated. If it evaluates to true, loopbody executes. Thereafter loopbody continues to execute as long as booleanexpression is true The for loop takes the form: for var := start to limit do loopbody var is an integer variable. The variable var is set to the value specified by start. If var limit, loopbody executes. After executing the loopbody, var is incremented by 1. The statement continues to execute until var > limit Discrete Mathematical Structures: Theory and Applications 75 Algorithms Pseudocode Conventions Control Structures The do/while loop takes the form: do loopbody while booleanexpression The loopbody is executed first and then the booleanexpression is evaluated. The loopbody continues to execute as long as the booleanexpression is true Discrete Mathematical Structures: Theory and Applications 76

39 Algorithms Pseudocode Conventions Block of Statement To consider a set of statements a single statement, the statements are written between the words begin and end begin statement1 statement2... statementn; end Discrete Mathematical Structures: Theory and Applications 77 Algorithms Pseudocode Conventions Return Statement The return statement is used to return the value computed by the algorithm and it takes the following form: return expression; The value specified by expression is returned. In an algorithm, the execution of a return statement also terminates the algorithm Read and Print Statements read x; Read the next value and store it in the variable x print x; Output the value of x Discrete Mathematical Structures: Theory and Applications 78

40 Algorithms Pseudocode Conventions Arrays (List) A list is a set of elements of the same type The length of the list is the number of elements in the list L[1...n ]. L is an array of n components, indexed 1 to n. L[i ] denotes the i th element of L For data in tabular form, a two-dimensional array is used: M[1...m,1...n ] M is a two-dimensional array of m rows and n columns The rows are indexed 1 to m and the columns are indexed 1 to n M[i,j] denotes the (i,j) th element of M, that is, the element at the i th row and j th column position Discrete Mathematical Structures: Theory and Applications 79 Pseudocode Conventions Algorithms Subprograms (Procedures) In a programming language,an algorithm is implemented in the form of a subprogram, a.k.a. a subroutine or a module Two types of subprograms Functions Returns a unique value Procedure Other types of subprograms body of the function or procedure is enclosed between the words begin and end the execution of a return statement in a function terminates the function Discrete Mathematical Structures: Theory and Applications 80

41 Algorithms Pseudocode Conventions Comments In describing the steps of an algorithm, comments are included wherever necessary to clarify the steps Two types of comments: single-line and multi-line Single-line comments start anywhere in the line with the pair of symbols // Multi-line comments are enclosed between the pair of symbols /* and */ Specifies what the algorithm does, as well as the input and output Discrete Mathematical Structures: Theory and Applications 81 Polynomial Operations Polynomial in one variable is Algorithms a i are real (or complex) and n is a non-negative integer If p(x) = a 0 then p(x) is a constant polynomial If p(x) is a nonzero constant polynomial, then the degree of p(x) is defined to be 0 If p(x) is not constant and a n 0, then n is called the degree of p(x); that is, the degree of a non-constant polynomial is defined to be the exponent of the highest power of x The basic operations performed on polynomials are to add, subtract, multiply, and divide polynomials, and to evaluate a polynomial at a given point Discrete Mathematical Structures: Theory and Applications 82

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

HANDOUT AND SET THEORY. Ariyadi Wijaya

HANDOUT 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 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

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

Intro to Logic and Proofs

Intro 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 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

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

Logic and Proof. Aiichiro Nakano

Logic and Proof. Aiichiro Nakano Logic and Proof Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Department of Computer Science Department of Physics & Astronomy Department of Chemical Engineering & Materials Science

More information

n 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

n 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 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

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

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

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

CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)

CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1) CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of

More information

CSC 125 :: Final Exam May 3 & 5, 2010

CSC 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 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

PROPOSITIONAL CALCULUS

PROPOSITIONAL CALCULUS PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both. These are not propositions! Connectives and

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 1 Elementary Logic

Chapter 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 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

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

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

Topics in Logic and Proofs

Topics in Logic and Proofs Chapter 2 Topics in Logic and Proofs Some mathematical statements carry a logical value of being true or false, while some do not. For example, the statement 4 + 5 = 9 is true, whereas the statement 2

More information

2. The Logic of Compound Statements Summary. Aaron Tan August 2017

2. 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 information

CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS

CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS 1.1 - Logical Form and Logical Equivalence Definition. A statement or proposition is a sentence that is either true or false, but not both. ex. 1 + 2 = 3 IS a statement

More information

Logic Overview, I. and T T T T F F F T F F F F

Logic 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 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

software design & management Gachon University Chulyun Kim

software 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 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

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

Math 3336: Discrete Mathematics Practice Problems for Exam I

Math 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 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

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

Section 1.1 Propositions

Section 1.1 Propositions Set Theory & Logic Section 1.1 Propositions Fall, 2009 Section 1.1 Propositions In Chapter 1, our main goals are to prove sentences about numbers, equations or functions and to write the proofs. Definition.

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

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

Propositional Logic. Spring Propositional Logic Spring / 32

Propositional Logic. Spring Propositional Logic Spring / 32 Propositional Logic Spring 2016 Propositional Logic Spring 2016 1 / 32 Introduction Learning Outcomes for this Presentation Learning Outcomes... At the conclusion of this session, we will Define the elements

More information

2-4: The Use of Quantifiers

2-4: The Use of Quantifiers 2-4: The Use of Quantifiers The number x + 2 is an even integer is not a statement. When x is replaced by 1, 3 or 5 the resulting statement is false. However, when x is replaced by 2, 4 or 6 the resulting

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

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

1.1 Language and Logic

1.1 Language and Logic c Oksana Shatalov, Fall 2017 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

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

Handout on Logic, Axiomatic Methods, and Proofs MATH Spring David C. Royster UNC Charlotte

Handout on Logic, Axiomatic Methods, and Proofs MATH Spring David C. Royster UNC Charlotte Handout on Logic, Axiomatic Methods, and Proofs MATH 3181 001 Spring 1999 David C. Royster UNC Charlotte January 18, 1999 Chapter 1 Logic and the Axiomatic Method 1.1 Introduction Mathematicians use a

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

Logic and Propositional Calculus

Logic and Propositional Calculus CHAPTER 4 Logic and Propositional Calculus 4.1 INTRODUCTION Many algorithms and proofs use logical expressions such as: IF p THEN q or If p 1 AND p 2, THEN q 1 OR q 2 Therefore it is necessary to know

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

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 The Foundation: Logic and Proofs

1 The Foundation: Logic and Proofs 1 The Foundation: Logic and Proofs 1.1 Propositional Logic Propositions( 명제 ) a declarative sentence that is either true or false, but not both nor neither letters denoting propositions p, q, r, s, T:

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

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

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

Proofs. Example of an axiom in this system: Given two distinct points, there is exactly one line that contains them.

Proofs. Example of an axiom in this system: Given two distinct points, there is exactly one line that contains them. Proofs A mathematical system consists of axioms, definitions and undefined terms. An axiom is assumed true. Definitions are used to create new concepts in terms of existing ones. Undefined terms are only

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

Lecture 3. Logic Predicates and Quantified Statements Statements with Multiple Quantifiers. Introduction to Proofs. Reading (Epp s textbook)

Lecture 3. Logic Predicates and Quantified Statements Statements with Multiple Quantifiers. Introduction to Proofs. Reading (Epp s textbook) Lecture 3 Logic Predicates and Quantified Statements Statements with Multiple Quantifiers Reading (Epp s textbook) 3.1-3.3 Introduction to Proofs Reading (Epp s textbook) 4.1-4.2 1 Propositional Functions

More information

First order Logic ( Predicate Logic) and Methods of Proof

First order Logic ( Predicate Logic) and Methods of Proof First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating

More information

Logic and Proof. On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes!

Logic and Proof. On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! Logic and Proof On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! 26-Aug-2011 MA 341 001 2 Requirements for Proof 1. Mutual understanding

More information

Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012

Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012 March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions

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

Discrete Mathematics for M.Sc.CS

Discrete Mathematics for M.Sc.CS Discrete Mathematics Page 1 CONTENTS UNIT-1 : SETS & RELATIONS 4-11 1. Discrete Mathematics Introduction... 4 2. Sets Set Definition... 4 Representation of a Set... 4 Cardinality of a Set... 5 Types of

More information

1 The Foundation: Logic and Proofs

1 The Foundation: Logic and Proofs 1 The Foundation: Logic and Proofs 1.1 Propositional Logic Propositions( ) a declarative sentence that is either true or false, but not both nor neither letters denoting propostions p, q, r, s, T: true

More information

Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture 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 information

On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! 26-Aug-2011 MA

On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! 26-Aug-2011 MA Logic and Proof On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! 26-Aug-2011 MA 341 001 2 Requirements for Proof 1. Mutual understanding

More information

The Logic of Compound Statements cont.

The 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 information

Chapter 2: The Logic of Quantified Statements

Chapter 2: The Logic of Quantified Statements Chapter 2: The Logic of Quantified Statements Topics include 2.1, 2.2 Predicates and Quantified Statements, 2.3 Statements with Multiple Quantifiers, and 2.4 Arguments with Quantified Statements. cs1231y

More information

Chapter 1, Logic and Proofs (3) 1.6. Rules of Inference

Chapter 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 information

Readings: Conjecture. Theorem. Rosen Section 1.5

Readings: Conjecture. Theorem. Rosen Section 1.5 Readings: Conjecture Theorem Lemma Lemma Step 1 Step 2 Step 3 : Step n-1 Step n a rule of inference an axiom a rule of inference Rosen Section 1.5 Provide justification of the steps used to show that a

More information

WUCT121. Discrete Mathematics. Logic. Tutorial Exercises

WUCT121. 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 information

Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.

Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms)

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

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

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

3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations

3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations Logic Propositions and logical operations Main concepts: propositions truth values propositional variables logical operations 1 Propositions and logical operations A proposition is the most basic element

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

1. SET 10/9/2013. Discrete Mathematics Fajrian Nur Adnan, M.CS

1. SET 10/9/2013. Discrete Mathematics Fajrian Nur Adnan, M.CS 1. SET 10/9/2013 Discrete Mathematics Fajrian Nur Adnan, M.CS 1 Discrete Mathematics 1. Set and Logic 2. Relation 3. Function 4. Induction 5. Boolean Algebra and Number Theory MID 6. Graf dan Tree/Pohon

More information

A. Propositional Logic

A. Propositional Logic CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals

More information

Chapter Summary. Sets (2.1) Set Operations (2.2) Functions (2.3) Sequences and Summations (2.4) Cardinality of Sets (2.5) Matrices (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. 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 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 Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs

More information

5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.

5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction. Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent

More 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

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

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

Logic and Set Notation

Logic and Set Notation Logic and Set Notation Logic Notation p, q, r: statements,,,, : logical operators p: not p p q: p and q p q: p or q p q: p implies q p q:p if and only if q We can build compound sentences using the above

More information

Basic properties of the Integers

Basic properties of the Integers Basic properties of the Integers Branko Ćurgus May 2, 2017 1 Axioms for the Integers In the axioms below we use the standard logical operators: conjunction, disjunction, exclusive disjunction, implication,

More information

Logic and Proofs. (A brief summary)

Logic and Proofs. (A brief summary) Logic and Proofs (A brief summary) Why Study Logic: To learn to prove claims/statements rigorously To be able to judge better the soundness and consistency of (others ) arguments To gain the foundations

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 Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs

More information

CSI30. Chapter 2. Basic Structures: Sets, Functions, Sequences, Sums. 2.1 Sets and subsets 2.2 Sets of sets

CSI30. Chapter 2. Basic Structures: Sets, Functions, Sequences, Sums. 2.1 Sets and subsets 2.2 Sets of sets Chapter 2. Basic Structures: Sets, Functions, Sequences, Sums 2.1 Sets and subsets 2.2 Sets of sets 1 Set is an unordered collection of objects. - used to group objects together, - often the objects with

More information

Sec$on Summary. Definition of sets Describing Sets

Sec$on Summary. Definition of sets Describing Sets Section 2.1 Sec$on Summary Definition of sets Describing Sets Roster Method Set-Builder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets

More information

1 Predicates and Quantifiers

1 Predicates and Quantifiers 1 Predicates and Quantifiers We have seen how to represent properties of objects. For example, B(x) may represent that x is a student at Bryn Mawr College. Here B stands for is a student at Bryn Mawr College

More information

Set Theory. CPT Section D Quantitative Aptitude Chapter 7 Brijeshwar Prasad Gupta

Set Theory. CPT Section D Quantitative Aptitude Chapter 7 Brijeshwar Prasad Gupta Set Theory CPT Section D Quantitative Aptitude Chapter 7 Brijeshwar Prasad Gupta Learning Objectives Number system Set Theory Set operations Product of Sets MCQ Number system Natural numbers:- N N = {1,2,3..}

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

ECOM Discrete Mathematics

ECOM Discrete Mathematics ECOM 2311- Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311- Discrete Mathematics - Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional

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

Logic and Propositional Calculus

Logic and Propositional Calculus CHAPTER 4 Logic and Propositional Calculus 4.1 INTRODUCTION Many algorithms and proofs use logical expressions such as: IF p THEN q or If p 1 AND p 2, THEN q 1 OR q 2 Therefore it is necessary to know

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

DISCRETE MATHEMATICS BA202

DISCRETE MATHEMATICS BA202 TOPIC 1 BASIC LOGIC This topic deals with propositional logic, logical connectives and truth tables and validity. Predicate logic, universal and existential quantification are discussed 1.1 PROPOSITION

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

Review 1. Andreas Klappenecker

Review 1. Andreas Klappenecker Review 1 Andreas Klappenecker 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,

More information

Logic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes:

Logic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes: Logic Logic is a discipline that studies the principles and methods used in correct reasoning It includes: A formal language for expressing statements. An inference mechanism (a collection of rules) to

More information

Propositional Logic Not Enough

Propositional Logic Not Enough Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks

More information