DISCRETE COMPUTATIONAL STRUCTURES

DISCRETE COMPUTATIONAL STRUCTURES
CSE 2353 Spring 2007

CSE 2353 OUTLINE Sets 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 Discrete Mathematical Structures: Theory and Applications

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

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

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 Discrete Mathematical Structures: Theory and Applications

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

Sets Superset Proper Subset
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

Sets Set Equality Empty (Null) Set
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

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

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 H = { 4 }, |H| = 1, H is a singleton Discrete Mathematical Structures: Theory and Applications

Sets Power Set Universal 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 Discrete Mathematical Structures: Theory and Applications

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

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} Discrete Mathematical Structures: Theory and Applications

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

Sets Disjoint Sets Example:
If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y =  Discrete Mathematical Structures: Theory and Applications

Sets Discrete Mathematical Structures: Theory and Applications

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} Discrete Mathematical Structures: Theory and Applications

Sets Complement 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

Sets Discrete Mathematical Structures: Theory and Applications

Sets Ordered Pair Cartesian Product
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

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 Discrete Mathematical Structures: Theory and Applications

Computer Implementation of Set Operations
Bit Map File Operations Intersection Union Element of Difference Complement Power Set Discrete Mathematical Structures: Theory and Applications

Special “Sets” in CS Multiset Ordered Set
Discrete Mathematical Structures: Theory and Applications

CSE 2353 OUTLINE Logic Sets Proof Techniques Relations and Posets
Functions Counting Principles 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 Discrete Mathematical Structures: Theory and Applications

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

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

Mathematical Logic Truth value Negation Truth Table
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 Discrete Mathematical Structures: Theory and Applications

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: Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Disjunction Truth Table for 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: Discrete Mathematical Structures: Theory and Applications

Mathematical Logic “If p, then q””
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: Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Implication p  q :
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 Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Biimplication “p if and only if q”
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: Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Statement Formulas Definitions
Symbols p ,q ,r ,...,called statement variables Symbols ~, ^, v, →,and ↔ are called logical connectives A statement variable is a statement formula 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 Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Precedence of logical connectives is:
~ highest ^ second highest v third highest → fourth highest ↔ fifth highest Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Tautology Contradiction
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

Mathematical Logic Logically Implies Logically Equivalent
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 Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Discrete Mathematical Structures: Theory and Applications

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 of statements. The final statement, , is the conclusion, and the statements are the premises of the argument. An argument is logically valid if the statement formula is a tautology. Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Modus Ponens:
Modus Tollens : Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Disjunctive Syllogisms:
Hypothetical Syllogism: Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Dilemma:
Conjunctive Simplification: Discrete Mathematical Structures: Theory and Applications

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

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

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” Two-place predicate: Discrete Mathematical Structures: Theory and Applications

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” Bound Variable The variable appearing in: or Discrete Mathematical Structures: Theory and Applications

Negation of Predicates (DeMorgan’s Laws) 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 , 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: and so, Discrete Mathematical Structures: Theory and Applications

Negation of Predicates (DeMorgan’s Laws) Discrete Mathematical Structures: Theory and Applications

Logic and CS Logic is basis of ALU Logic is crucial to IF statements
OR NOT Implementation of quantifiers Looping Database Query Languages Relational Algebra Relational Calculus SQL Discrete Mathematical Structures: Theory and Applications

Integers and Inductions
CSE OUTLINE Sets Logic Proof Techniques Integers and Inductions Relations and Posets Functions Counting Principles Boolean Algebra

Proof Technique: Learning Objectives
Learn various proof techniques Direct Indirect Contradiction Induction Practice writing proofs CS: Why study proof techniques? Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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

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 Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

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

Proof of Equivalent Statements
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? Discrete Mathematical Structures: Theory and Applications

Other Proof Techniques
Vacuous Trivial Contrapositive Counter Example Divide into Cases Constructive Discrete Mathematical Structures: Theory and Applications

You can not prove by example
Proof Basics You can not prove by example Discrete Mathematical Structures: Theory and Applications

Proofs in Computer Science
Proof of program correctness Proofs are used to verify approaches Discrete Mathematical Structures: Theory and Applications

Integers and Induction
CSE OUTLINE Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

Learning Objectives Learn about the basic properties of integers
Explore how addition and subtraction operations are performed on binary numbers Learn how the principle of mathematical induction is used to solve problems CS Become aware how integers are represented in computer memory Looping Discrete Mathematical Structures: Theory and Applications

Integers Properties of Integers

Integers Discrete Mathematical Structures: Theory and Applications

The div and mod operators
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 Discrete Mathematical Structures: Theory and Applications

Integers Discrete Mathematical Structures: Theory and Applications

Integers Relatively Prime Number

Integers Least Common Multiples

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Two’s Complements and Operations on Binary Numbers In computer memory, integers are represented as binary numbers in fixed-length 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 Discrete Mathematical Structures: Theory and Applications

One’s Complements and Operations on Binary Numbers Discrete Mathematical Structures: Theory and Applications

Mathematical Deduction

Proof of a mathematical statement by the principle of mathematical induction consists of three steps: Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Prove that P(k+1) is true, i.e. Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Suppose some P(k) is true, i.e. the kth staircase is climbed for some By the assumption, because the kth staircase was climbed, the k+1st staircase was climbed Therefore, P(k) is true, so Discrete Mathematical Structures: Theory and Applications

We can associate a predicate, P(n). The predicate P(n) is such that: Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

Prime Numbers Discrete Mathematical Structures: Theory and Applications

Factoring a Positive Integer
Prime Numbers Factoring a Positive Integer The standard factorization of n Discrete Mathematical Structures: Theory and Applications

Fermat’s Factoring Method
Prime Numbers Fermat’s Factoring Method Discrete Mathematical Structures: Theory and Applications

Learn about relations and their basic properties
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 Discrete Mathematical Structures: Theory and Applications

Relations are a natural way to associate objects of various sets

Relations R can be described in Roster form Set-builder form

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 Discrete Mathematical Structures: Theory and Applications

Relations Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Domain and Range of the Relation
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 Discrete Mathematical Structures: Theory and Applications

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) Discrete Mathematical Structures: Theory and Applications

Relations Constructing New Relations from Existing Relations

Relations Example: Consider the relations R and S as given in Figure 3.7. The composition S ◦ R is given by Figure 3.8. Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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) Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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) Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

Explore various properties of functions Learn about binary operations
Learning Objectives Learn about functions Explore various properties of functions Learn about binary operations Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

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: 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 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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

The image of g is Im(g) = {a, b, c , d} = B
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 Discrete Mathematical Structures: Theory and Applications

Functions Example 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 one-one. 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-to-one correspondence. Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4} and B = {a, b, c , d, e}
Example Let A = {1,2,3,4} and B = {a, b, c , d, e} f : 1 → a, 2 → a, 3 → a, → 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. Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4} and B = {a, b, c , d, e}
f : 1 → a, 2 → b, 3 → d, → 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. Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set

Binary Operations Discrete Mathematical Structures: Theory and Applications

Learn the basic counting principles— multiplication and addition
Learning Objectives Learn the basic counting principles— multiplication and addition Explore the pigeonhole principle Learn about permutations Learn about combinations Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles

Pigeonhole Principle The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle. Discrete Mathematical Structures: Theory and Applications

Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

Permutations Discrete Mathematical Structures: Theory and Applications

Combinations Discrete Mathematical Structures: Theory and Applications

Generalized Permutations and Combinations

Two-Element Boolean Algebra
Let B = {0, 1}. Discrete Mathematical Structures: Theory and Applications

Two-Element Boolean Algebra

Boolean Algebra Discrete Mathematical Structures: Theory and Applications

Logical Gates and Combinatorial Circuits

Logical Gates and Combinatorial Circuits
The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression. Discrete Mathematical Structures: Theory and Applications

DISCRETE COMPUTATIONAL STRUCTURES

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DISCRETE COMPUTATIONAL STRUCTURES

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