DISCRETE COMPUTATIONAL STRUCTURES

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DISCRETE COMPUTATIONAL STRUCTURES

• CSE 2353
• Material for Second Test
• Spring 2006

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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

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Proof Technique Learning Objectives

• Learn various proof techniques
• Direct
• Indirect
• Contradiction
• Induction
• Practice writing proofs
• CS Why study proof techniques?

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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

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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

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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

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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 p1p2pn1. 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

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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

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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

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Other Proof Techniques

• Vacuous
• Trivial
• Contrapositive
• Counter Example
• Divide into Cases

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Proof Basics

• You can not prove by example

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Proofs in Computer Science

• Proof of program correctness
• Proofs are used to verify approaches

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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

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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

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Integers

• Properties of Integers

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Integers

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Integers

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Integers
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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

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Integers
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Integers
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Integers
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Integers

• Relatively Prime Number

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Integers

• Least Common Multiples

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Representation of Integers in Computer

• Electrical signals are used inside the computer
  to process information
• Two types of signals
• Analog
• Continuous wave forms used to represent such
  things as sound
• Examples audio tapes, older television signals,
  etc.
• Digital
• Represent information with a sequence of 0s and
  1s
• Examples compact discs, newer digital HDTV
  signals

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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

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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

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Representation of Integers in Computers
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Representation of Integers in Computers

• Twos 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

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Representation of Integers in Computers
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Representation of Integers in Computers

• Ones Complements and Operations on Binary
  Numbers

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Representation of Integers in Computers
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Mathematical Deduction
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Mathematical Deduction

• Proof of a mathematical statement by the
  principle of mathematical induction consists of
  three steps

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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

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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
• Prove that P(k1) is true, i.e.

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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

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Mathematical Deduction

• 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 k1st staircase was climbed
• Therefore, P(k) is true, so

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Mathematical Deduction
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Mathematical Deduction

• We can associate a predicate, P(n). The predicate
  P(n) is such that

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Prime Numbers
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Prime Numbers
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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.
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Prime Numbers
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Prime Numbers

• Factoring a Positive Integer
• The standard factorization of n

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Prime Numbers

• Fermats Factoring Method

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Prime Numbers

• Fermats Factoring Method

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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

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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

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Relations

• Relations are a natural way to associate objects
  of various sets

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Relations

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

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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

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Relations
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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

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Relations
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Relations

• Directed graph (Digraph) representation of R
• Each dot is called a vertex
• If a vertex is labeled, a, then it is also called
  vertex a
• An arc from a vertex labeled a, to another
  vertex, b is called a directed edge, or directed
  arc from a to b
• The ordered pair (A , R) a directed graph, or
  digraph, of the relation R, where each element of
  A is a called a vertex of the digraph

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Relations

• Directed graph (Digraph) representation of R
  (Continued)
• For vertices a and b , if a R b, a is adjacent to
  b and b is adjacent from a
• Because (a, a) ? R, an arc from a to a is drawn
  because (a, b) ? R, an arc is drawn from a to b.
  Similarly, arcs are drawn from b to b, b to c , b
  to a, b to d, and c to d
• For an element a ? A such that (a, a) ? R, a
  directed edge is drawn from a to a. Such a
  directed edge is called a loop at vertex a

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Relations

• Directed graph (Digraph) representation of R
  (Continued)
• Position of each vertex is not important
• In the digraph of a relation R, there is a
  directed edge or arc from a vertex a to a vertex
  b if and only if a R b
• Let A a ,b ,c ,d and let R be the relation
  defined by the following set
• R (a ,a ), (a ,b ), (b ,b ), (b ,c ), (b
  ,a ), (b ,d ), (c ,d )

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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

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Relations
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Relations
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Relations
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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)

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Relations
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Relations

• Constructing New Relations from Existing
  Relations

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Relations

• Example
• Consider the relations R and S as given in Figure
  3.7.
• The composition S ? R is given by Figure 3.8.

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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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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

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Partially Ordered Sets
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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)

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Partially Ordered Sets
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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
• In this poset, there is no element b ? S such
  that b ? 5 and b divides 5. (That is, 5 is not
  divisible by any other element of S except 5).
  Hence, 5 is a minimal element. Similarly, 2 is a
  minimal element

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Partially Ordered Sets

• Hasse Diagram
• 10 is not a minimal element because 2 ? S and 2
  divides 10. That is, there exists an element b ?
  S such that b lt 10. Similarly, 4, 15, and 20 are
  not minimal elements
• 2 and 5 are the only minimal elements of this
  poset. Notice that 2 does not divide 5.
  Therefore, it is not true that 2 b, for all b ?
  S, and so 2 is not a least element in (S,).
  Similarly, 5 is not a least element. This poset
  has no least element

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Partially Ordered Sets
Figure 3.24

• Hasse Diagram
• There is no element b ? S such that b ?15, b gt
  15, and 15 divides b. That is, there is no
  element b ? S such that 15 lt b. Thus, 15 is a
  maximal element. Similarly, 20 is a maximal
  element.
• 10 is not a maximal element because 20 ? S and 10
  divides 20. That is, there exists an element b ?
  S such that 10 lt b. Similarly, 4 is not a maximal
  element.

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Partially Ordered Sets
Figure 3.24

• Hasse Diagram
• 20 and 15 are the only maximal elements of this
  poset
• 10 does not divide 15, hence it is not true that
  b 15, for all b ? S, and so 15 is not a
  greatest element in (S,)
• This poset has no greatest element

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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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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

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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

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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

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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)

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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.