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DISCRETE COMPUTATIONAL STRUCTURES CSE 2353 Fall 2011 Most slides modified from Discrete Mathematical Structures: Theory and Applications CSE 2353 OUTLINE 1. 2. 3. 4. 5. 6. 7. 8. Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra CSE 2353 OUTLINE 1.Sets 2. 3. 4. 5. 6. 7. 8. Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra Sets: Learning Objectives Learn about sets Explore various operations on sets Become familiar with Venn diagrams CS: Learn how to represent sets in computer memory Learn how to implement set operations in programs CSE 2353 f11 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 CSE 2353 f11 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} CSE 2353 f11 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 positive 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 CSE 2353 f11 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 CSE 2353 f11 Z, since a Y, but a Z Sets Superset 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 CSE 2353 f11 Sets Set Equality 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 CSE 2353 f11 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 CSE 2353 f11 Sets Cardinality of 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 CSE 2353 f11 Sets Power Set 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 CSE 2353 f11 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 CSE 2353 f11 Set Operations and Venn Diagrams Union of Sets Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then XUY = {1,2,3,4,5,6,7,8,9} CSE 2353 f11 Sets Intersection of Sets Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X ∩ Y = {5} CSE 2353 f11 Sets Disjoint Sets Example: If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y = CSE 2353 f11 Sets CSE 2353 f11 Sets CSE 2353 f11 Sets Difference • Example: If X = {a,b,c,d} and Y = {c,d,e,f}, then X – Y = {a,b} and Y – X = {e,f} CSE 2353 f11 Sets Complement Example: If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then X’ = {a,b} CSE 2353 f11 Sets CSE 2353 f11 Sets CSE 2353 f11 Sets CSE 2353 f11 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)} CSE 2353 f11 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. Linked List Solution: use Bit Strings (Bit Map) 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 A set may be implemented as a file CSE 2353 f11 Computer Implementation of Set Operations Bit Map File Operations Intersection Union Element of Difference Complement Power Set CSE 2353 f11 Special “Sets” in CS Multiset Ordered Set CSE 2353 f11 CSE 2353 OUTLINE 1. Sets 2.Logic 3. Proof Techniques 4. Relations and Posets 5. Functions 6. Counting Principles 7. Boolean Algebra Logic: Learning Objectives Learn about statements (propositions) Learn how to use logical connectives to combine statements Explore how to draw conclusions using various argument forms Become familiar with quantifiers and predicates CS Boolean data type If statement Impact of negations Implementation of quantifiers CSE 2353 f11 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 CSE 2353 f11 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? CSE 2353 f11 Mathematical Logic Truth value One of the values “truth” (T) or “falsity” (F) assigned to a statement Negation The negation of p, written ~p, is the statement obtained by negating statement p Example: p: A is a consonant ~p: it is the case that A is not a consonant Truth Table CSE 2353 f11 Mathematical Logic 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 Truth Table for Conjunction: CSE 2353 f11 Mathematical Logic Disjunction Let p and q be statements. The disjunction of p and q, written p v q , is the statement formed by joining statements p and q using the word “or” The statement p v q is true if at least one of the statements p and q is true; otherwise p v q is false The symbol v is read “or” Truth Table for Disjunction: CSE 2353 f11 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 “If p, then q”” p is called the hypothesis, q is called the conclusion Truth Table for Implication: CSE 2353 f11 Mathematical Logic Implication Let p: Today is Sunday and q: I will wash the car. 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 CSE 2353 f11 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 if and only if q” Truth Table for the Biconditional: CSE 2353 f11 Mathematical Logic Statement Formulas Definitions Symbols p ,q ,r ,...,called statement variables Symbols ~, , v, →,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 v B ), (A → B ) ^ and (A ↔ B ) are statement formulas Expressions are statement formulas that are constructed only by using 1) and 2) above CSE 2353 f11 Mathematical Logic Precedence of logical connectives is: ~ highest ^ second highest v third highest → fourth highest ↔ fifth highest CSE 2353 f11 Mathematical Logic Tautology 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 CSE 2353 f11 Mathematical Logic Logically Implies 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 CSE 2353 f11 Mathematical Logic CSE 2353 f11 Validity of Arguments Proof: an argument or a proof of a theorem consists of a finite sequence of statements ending in a conclusion Argument: a finite sequence A1 , A2 , A3 , ..., An1 , An of statements. The final statement, An , is the conclusion, and the statements A1 , A2 , A3 , ..., An 1 are the premises of the argument. An argument is logically valid if the statement formula A1 , A2 , A3 , ..., An1 An is a tautology. CSE 2353 f11 Validity of Arguments Valid Argument Forms Modus Ponens: Modus Tollens : CSE 2353 f11 Validity of Arguments Valid Argument Forms Disjunctive Syllogisms: Hypothetical Syllogism: CSE 2353 f11 Validity of Arguments Valid Argument Forms Dilemma: Conjunctive Simplification: CSE 2353 f11 Validity of Arguments Valid Argument Forms Disjunctive Addition: Conjunctive Addition: CSE 2353 f11 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 CSE 2353 f11 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) or For every x, P(x) The symbol is read as “for all and every” x P ( x ) Two-place predicate: xy P( x, y ) CSE 2353 f11 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 is read as “there exists” x P ( x ) Bound Variable The variable appearing in: CSE 2353 f11 x P ( x) or x P ( x ) 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 ( x) and so, ~ x P( x) x ~ P( x) CSE 2353 f11 Quantifiers and First Order Logic Negation of Predicates (DeMorgan’s Laws) ~ x P( x) x ~ P( x) CSE 2353 f11 Arguments in Predicate Logic Universal Specification If x F ( x) is true, then a U F(a) is true Universal Generalization If F(a) is true a U then x F ( x) is true Existential Specification If x F ( x ) is true, then a U where F(a) is true Existential Generalization If F(a) is true a U then x F ( x ) is true CSE 2353 f11 54 Logic and CS Logic is basis of ALU Logic is crucial to IF statements AND OR NOT Implementation of quantifiers Looping Database Query Languages Relational Algebra Relational Calculus SQL CSE 2353 f11 CSE 2353 OUTLINE 1. Sets 2. Logic 3. Proof Techniques 4. Integers and Inductions 5. Relations and Posets 6. Functions 7. Counting Principles 8. Boolean Algebra Proof Technique: Learning Objectives Learn various proof techniques Direct Indirect Contradiction Induction Practice writing proofs CS: Why study proof techniques? CSE 2353 f11 Proof Techniques Theorem Statement that can be shown to be true (under certain conditions) Typically Stated in one of three ways As Facts As Implications As Biimplications CSE 2353 f11 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 CSE 2353 f11 Proof Techniques Indirect Proof 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 CSE 2353 f11 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. p1,p2,…,pn Consider the number q = p1p2…pn+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. CSE 2353 f11 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 CSE 2353 f11 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 What other methods are possible? CSE 2353 f11 Other Proof Techniques Vacuous Trivial Contrapositive Counter Example Divide into Cases Constructive CSE 2353 f11 Proof Basics You can not prove by example CSE 2353 f11 Proofs in Computer Science Proof of program correctness Proofs are used to verify approaches CSE 2353 f11 CSE 2353 OUTLINE 1. Sets 2. Logic 3. Proof Techniques 4. Integers and Induction 5. Relations and Posets 6. Functions 7. Counting Principles 8. Boolean Algebra Learning Objectives Learn how the principle of mathematical induction is used to solve problems and proofs Learn about the basic properties of integers Explore how addition and subtraction operations are performed on binary numbers CS Become aware how integers are represented in computer memory Looping CSE 2353 f11 Mathematical Deduction CSE 2353 f11 Mathematical Deduction Proof of a mathematical statement by the principle of mathematical induction consists of three steps: CSE 2353 f11 Mathematical Deduction Assume that when a domino is knocked over, the next domino is knocked over by it Show that if the first domino is knocked over, then all the dominoes will be knocked over CSE 2353 f11 Mathematical Deduction Let P(n) denote the statement that then nth domino is knocked over Show that P(1) is true Assume some P(k) is true, i.e. the kth domino is knocked over for some k 1 Prove that P(k+1) is true, i.e. P ( k ) P ( k 1) CSE 2353 f11 Mathematical Deduction Assume that when a staircase is climbed, the next staircase is also climbed Show that if the first staircase is climbed then all staircases can be climbed Let P(n) denote the statement that then nth staircase is climbed It is given that the first staircase is climbed, so P(1) is true CSE 2353 f11 Mathematical Deduction Suppose some P(k) is true, i.e. the kth staircase is climbed for some k 1 By the assumption, because the kth staircase was climbed, the k+1st staircase was climbed Therefore, P(k) is true, so P ( k ) P ( k 1) CSE 2353 f11 Mathematical Deduction CSE 2353 f11 Mathematical Deduction We can associate a predicate, P(n). The predicate P(n) is such that: CSE 2353 f11 Integers Properties of Integers CSE 2353 f11 Integers CSE 2353 f11 Integers The div and mod operators div a div b = the quotient of a and b obtained by dividing a on b. Examples: 8 div 5 = 1 13 div 3 = 4 mod a mod b = the remainder of a and b obtained by dividing a on b 8 mod 5 = 3 13 mod 3 = 1 CSE 2353 f11 Integers CSE 2353 f11 Integers CSE 2353 f11 Integers Relatively Prime Number CSE 2353 f11 Integers Least Common Multiples CSE 2353 f11 Representation of Integers in Computers Digital Signals 0s and 1s – 0s represent low voltage, 1s high voltage Digital signals are more reliable carriers of information than analog signals Can be copied from one device to another with exact precision Machine language is a sequence of 0s and 1s The digit 0 or 1 is called a binary digit , or bit A sequence of 0s and 1s is sometimes referred to as binary code CSE 2353 f11 Representation of Integers in Computers Decimal System or Base-10 The digits that are used to represent numbers in base 10 are 0,1,2,3,4,5,6,7,8, and 9 Binary System or Base-2 Computer memory stores numbers in machine language, i.e., as a sequence of 0s and 1s Octal System or Base-8 Digits that are used to represent numbers in base 8 are 0,1,2,3,4,5,6, and 7 Hexadecimal System or Base-16 Digits and letters that are used to represent numbers in base 16 are 0,1,2,3,4,5,6,7,8,9,A ,B ,C ,D ,E , and F CSE 2353 f11 Representation of Integers in Computers CSE 2353 f11 Representation of Integers in Computers Two’s Complements and Operations on Binary Numbers In computer memory, integers are represented as binary numbers in fixedlength bit strings, such as 8, 16, 32 and 64 Assume that integers are represented as 8-bit fixed-length strings Sign bit is the MSB (Most Significant Bit) Leftmost bit (MSB) = 0, number is positive Leftmost bit (MSB) = 1, number is negative CSE 2353 f11 Representation of Integers in Computers CSE 2353 f11 Representation of Integers in Computers One’s Complements and Operations on Binary Numbers CSE 2353 f11 Representation of Integers in Computers CSE 2353 f11 Prime Numbers Example: Consider the integer 131. Observe that 2 does not divide 131. We now find all odd primes p such that p2 131. These primes are 3, 5, 7, and 11. Now none of 3, 5, 7, and 11 divides 131. Hence, 131 is a prime. CSE 2353 f11 Prime Numbers CSE 2353 f11 Prime Numbers Factoring a Positive Integer The standard factorization of n CSE 2353 f11 Prime Numbers Fermat’s Factoring Method CSE 2353 f11 Prime Numbers Fermat’s Factoring Method CSE 2353 f11 CSE 2353 OUTLINE 1. Sets 2. Logic 3. Proof Techniques 4. Integers and Induction 5.Relations and Posets 6. Functions 7. Counting Principles 8. Boolean Algebra Learning Objectives Learn about relations and their basic properties Explore equivalence relations Become aware of closures Learn about posets Explore how relations are used in the design of relational databases CSE 2353 f11 Relations Relations are a natural way to associate objects of various sets CSE 2353 f11 Relations R can be described in Roster form Set-builder form CSE 2353 f11 Relations Arrow Diagram Write the elements of A in one column Write the elements B in another column Draw an arrow from an element, a, of A to an element, b, of B, if (a ,b) R Here, A = {2,3,5} and B = {7,10,12,30} and R from A into B is defined as follows: For all a A and b B, a R b if and only if a divides b The symbol → (called an arrow) represents the relation R CSE 2353 f11 Relations CSE 2353 f11 Relations Directed Graph Let R be a relation on a finite set A Describe R pictorially as follows: For each element of A , draw a small or big dot and label the dot by the corresponding element of A Draw an arrow from a dot labeled a , to another dot labeled, b , if a R b . Resulting pictorial representation of R is called the directed graph representation of the relation R CSE 2353 f11 Relations CSE 2353 f11 Relations Domain and Range of the Relation Let R be a relation from a set A into a set B. Then R ⊆ A x B. The elements of the relation R tell which element of A is R-related to which element of B CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Relations Let A = {1, 2, 3, 4} and B = {p, q, r}. Let R = {(1, q), (2, r ), (3, q), (4, p)}. Then R−1 = {(q, 1), (r , 2), (q, 3), (p, 4)} To find R−1, just reverse the directions of the arrows D(R) = {1, 2, 3, 4} = Im(R−1), Im(R) = {p, q, r} = D(R−1) CSE 2353 f11 Relations CSE 2353 f11 Relations Constructing New Relations from Existing Relations CSE 2353 f11 Relations Example: Consider the relations R and S as given in Figure 3.7. The composition S ◦ R is given by Figure 3.8. CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Relations CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets Hasse Diagram Let S = {1, 2, 3}. Then P(S) = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S} Now (P(S),≤) is a poset, where ≤ denotes the set inclusion relation. The poset diagram of (P(S),≤) is shown in Figure 3.22 CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets Hasse Diagram Let S = {1, 2, 3}. Then P(S) = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S} (P(S),≤) is a poset, where ≤ denotes the set inclusion relation Draw the digraph of this inclusion relation (see Figure 3.23). Place the vertex A above vertex B if B ⊂ A. Now follow steps (2), (3), and (4) CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets Hasse Diagram Consider the poset (S,≤), where S = {2, 4, 5, 10, 15, 20} and the partial order ≤ is the divisibility relation. 2 and 5 are the only minimal elements of this poset. This poset has no least element. 20 and 15 are the only maximal elements of this poset. This poset has no greatest element. CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Partially Ordered Sets CSE 2353 f11 Application: Relational Database A database is a shared and integrated computer structure that stores End-user data; i.e., raw facts that are of interest to the end user; Metadata, i.e., data about data through which data are integrated A database can be thought of as a well-organized electronic file cabinet whose contents are managed by software known as a database management system; that is, a collection of programs to manage the data and control the accessibility of the data CSE 2353 f11 Application: Relational Database In a relational database system, tables are considered as relations A table is an n-ary relation, where n is the number of columns in the tables The headings of the columns of a table are called attributes, or fields, and each row is called a record The domain of a field is the set of all (possible) elements in that column CSE 2353 f11 Application: Relational Database Each entry in the ID column uniquely identifies the row containing that ID Such a field is called a primary key Sometimes, a primary key may consist of more than one field CSE 2353 f11 Application: Relational Database Structured Query Language (SQL) Information from a database is retrieved via a query, which is a request to the database for some information A relational database management system provides a standard language, called structured query language (SQL) A relational database management system provides a standard language, called structured query language (SQL) CSE 2353 f11 Application: Relational Database Structured Query Language (SQL) An SQL contains commands to create tables, insert data into tables, update tables, delete tables, etc. Once the tables are created, commands can be used to manipulate data into those tables. The most commonly used command for this purpose is the select command. The select command allows the user to do the following: Specify what information is to be retrieved and from which tables. Specify conditions to retrieve the data in a specific form. Specify how the retrieved data are to be displayed. CSE 2353 f11 CSE 2353 OUTLINE 1. Sets 2. Logic 3. Proof Techniques 4. Integers and Induction 5. Relations and Posets 6.Functions 7. Counting Principles 8. Boolean Algebra Learning Objectives Learn about functions Explore various properties of functions Learn about binary operations CSE 2353 f11 Functions CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 Functions Every function is a relation Therefore, functions on finite sets can be described by arrow diagrams. In the case of functions, the arrow diagram may be drawn slightly differently. If f : A → B is a function from a finite set A into a finite set B, then in the arrow diagram, the elements of A are enclosed in ellipses rather than individual boxes. CSE 2353 f11 Functions To determine from its arrow diagram whether a relation f from a set A into a set B is a function, two things are checked: 1) Check to see if there is an arrow from each element of A to an element of B This would ensure that the domain of f is the set A, i.e., D(f) = A 2) Check to see that there is only one arrow from each element of A to an element of B This would ensure that f is well defined CSE 2353 f11 Functions Let A = {1,2,3,4} and B = {a, b, c , d} be sets The arrow diagram in Figure 5.6 represents the relation f from A into B Every element of A has some image in B An element of A is related to only one element of B; i.e., for each a ∈ A there exists a unique element b ∈ B such that f (a) = b CSE 2353 f11 Functions Therefore, f is a function from A into B The image of f is the set Im(f) = {a, b, d} There is an arrow originating from each element of A to an element of B D(f) = A There is only one arrow from each element of A to an element of B f is well defined CSE 2353 f11 Functions The arrow diagram in Figure 5.7 represents the relation g from A into B Every element of A has some image in B D(g ) = A For each a ∈ A, there exists a unique element b ∈ B such that g(a) = b g is a function from A into B CSE 2353 f11 Functions The image of g is Im(g) = {a, b, c , d} =B There is only one arrow from each element of A to an element of B g is well defined CSE 2353 f11 Functions CSE 2353 f11 Functions CSE 2353 f11 Functions Example 5.1.16 CSE 2353 f11 Let A = {1,2,3,4} and B = {a, b, c , d}. Let f : A → B be a function such that the arrow diagram of f is as shown in Figure 5.10 The arrows from a distinct element of A go to a distinct element of B. That is, every element of B has at most one arrow coming to it. If a1, a2 ∈ A and a1 = a2, then f(a1) = f(a2). Hence, f is oneone. Each element of B has an arrow coming to it. That is, each element of B has a preimage. Im(f) = B. Hence, f is onto B. It also follows that f is a one-toone correspondence. Functions Example 5.1.18 CSE 2353 f11 Let A = {1,2,3,4} and B = {a , b , c , d , e } f : 1 → a, 2 → a, 3 → a, 4 →a For this function the images of distinct elements of the domain are not distinct. For example 1 2, but f(1) = a = f(2) . Im(f) = {a} B. Hence, f is neither one-one nor onto B. Functions Let A = {1,2,3,4} and B = {a, b, c , d, e } f : 1 → a, 2 → b, 3 → d, 4→e f is one-one. In this function, for the element c of B, the codomain, there is no element x in the domain such that f(x) = c ; i.e., c has no preimage. Hence, f is not onto B. CSE 2353 f11 Functions CSE 2353 f11 Functions Let A = {1,2,3,4}, B = {a, b, c , d, e},and C = {7,8,9}. Consider the functions f : A → B, g : B → C as defined by the arrow diagrams in Figure 5.14. The arrow diagram in Figure 5.15 describes the function h = g ◦ f : A → C. CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 CSE 2353 f11 Special Functions and Cardinality of a Set CSE 2353 f11 Binary Operations CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 OUTLINE 1. Sets 2. Logic 3. Proof Techniques 4. Integers and Induction 5. Relations and Posets 6. Functions 7.Counting Principles 8. Boolean Algebra Learning Objectives Learn the basic counting principles— multiplication and addition Explore the pigeonhole principle Learn about permutations Learn about combinations CSE 2353 f11 Basic Counting Principles CSE 2353 f11 Basic Counting Principles CSE 2353 f11 Pigeonhole Principle The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle. CSE 2353 f11 Pigeonhole Principle CSE 2353 f11 Permutations CSE 2353 f11 Permutations CSE 2353 f11 Combinations CSE 2353 f11 Combinations CSE 2353 f11 Generalized Permutations and Combinations CSE 2353 f11 CSE 2353 OUTLINE 1. Sets 2. Logic 3. Proof Techniques 4. Integers and Induction 5. Relations and Posets 6. Functions 7. Counting Principles 8.Boolean Algebra Two-Element Boolean Algebra Let B = {0, 1}. CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 Two-Element Boolean Algebra CSE 2353 f11 Two-Element Boolean Algebra CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 Boolean Algebra CSE 2353 f11 Boolean Algebra CSE 2353 f11 Logical Gates and Combinatorial Circuits CSE 2353 f11 Logical Gates and Combinatorial Circuits CSE 2353 f11 Logical Gates and Combinatorial Circuits CSE 2353 f11 Logical Gates and Combinatorial Circuits CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 Logical Gates and Combinatorial Circuits The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression. CSE 2353 f11 CSE 2353 f11 CSE 2353 f11 CSE 2353 f11