461191 Discrete Math Lecture 5: Counting

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461191 Discrete MathLecture 5 Counting

• San Ratanasanya
• CS, KMUTNB

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Todays topics

• Review of Induction and Recursion
• Administrivia
• Basics of Counting
• Prolog Tutorial
• Python Tutorial

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Induction

• Based on the Rule of Inference (or Well-Ordering)
• It has two steps basis step and inductive step
• If P(1) and ?k(P(k) ? P(k1)) are true for
  positive integers then ?P(n) is true
• Can be used to prove tremendous variety of
  results
• Strong induction is more flexible proof
  technique. It shows P(1)?P(2) ? ? P(k) ?
  P(k1) is true in the inductive step instead of
  just showing only P(k) ? P(k1) as in
  mathematical induction

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Example

• Prove this theorem All horse are the same
  color
• Proof
• Basis Step no horse ? vacuously true!
• Inductive Step assume k horse have the same
  color (inductive hypothesis) is true.
• Then we have to show that k1 horse also have the
  same color.

Second set of k horse having the same color
k1 horse having the same color
First set of k horse having the same color
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Example

• For any 2n x 2n board, we can chase Bill to the
  corner.
• Proof
• Basis Step no board (n0), no need to need to
  chase Bill with an L-shape tile.
• Inductive Step assume we can get Bill to the
  corner for 2n x 2n board.
• We need to show it is true for 2n1 x 2n1

We are done!
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Recursion

• Another way to define function based on induction
• Specifies some initial terms and define rules for
  finding subsequent values from values already
  known Recursive Definition
• Use induction to prove recursive definition
• Algorithms can also be recursively designed

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Example

• Give a recursive definition of

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Program Verification

• A method to verify that a designed procedure or
  program always produce the correct answer
• The verification is based on rules of inference
• Initial assertion and Final assertion
• properties that the input must have
• The properties that output should have
• Partial correctness vs. Total correctness
• The correct answer is obtained if program
  terminates
• The program always terminated

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Example
procedure multiply (m, n integers) if n lt 0 then
a -n else a n k 0 x 0
while k lt a begin x x m k k 1 end
S1
S3
S2
if n lt 0 then product -x else product x
S4

• Prove that this program is correct.

Let p m and n are integers and q p ? (an).
Then we can prove that pS1q is true. Let r q
? (k0) ? (x0). Then it can be proved that
qS2r is true. S4 can be proved in the same way
as S1. Now, we have to show S3 is
correct. First, we find loop invariant in S3
which is s xm?k ? k ? a or xm?a ? an.
If r is true, we know that s is true before
entering the loop. Moreover, this loop terminates
when k a and x ma. That is partial
correctness is held. It follows that rS3s is
true. Now that we prove S1, S2, S3, and S4 are
true so, by composition rule we could show
that pS1q ? qS2r ? rS3s ? tS4u ?
pS1S2S3S4u is true. ?pSu is correct.
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Administrivia

• Midterm Exam is in next 3 weeks. Please be
  prepared!
• All assignments MUST be submitted before Midterm.

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Basics of Counting
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The Importance of Counting

• To count the number of solutions to solve the
  problem
• Determine the Complexity of Algorithms
• To Use Passwords for Computer Security

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Combinatorics

• Count the number of ways to put things together
  into various combinations
• e.g., If a password is 6-8 letters and/or digits,
  how many passwords can there be?
• Two main rules
• Sum
• Product

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Sum Rule

• Let us consider two tasks
• m is the number of ways to do task 1
• n is the number of ways to do task 2
• Tasks are independent of each other, i.e.,
• Performing task 1 does not accomplish task 2 and
  vice versa.
• Sum rule the number of ways that either task 1
  or task 2 can be done, but not both, is m n.
• Generalizes to multiple tasks ...

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Example

• A student can choose a computer project from one
  of three lists. The three lists contain 23, 15,
  and 19 possible projects respectively. How many
  possible projects are there to choose from?

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Set Theoretic Version

• If A is the set of ways to do task 1, and B the
  set of ways to do task 2, and if A and B are
  disjoint, then
• the ways to do either task 1 or 2 are
• A?B, and A?BAB

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Product Rule

• Let us consider two tasks
• m is the number of ways to do task 1
• n is the number of ways to do task 2
• Tasks are independent of each other, i.e.,
• Performing task 1does not accomplish task 2 and
  vice versa.
• Product rule the number of ways that both tasks
  1 and 2 can be done in mn.
• Generalizes to multiple tasks ...

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Example

• The chairs of an auditorium are to be labeled
  with a letter and a positive integer not to
  exceed 100. What is the largest number of chairs
  that can be labeled differently?

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Set Theoretic Version

• If A is the set of ways to do task 1, and B the
  set of ways to do task 2, and if A and B are
  disjoint, then
• The ways to do both task 1 and 2 can be
  represented as A?B, and A?BAB

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More Examples

• How many different bit strings are there of
  length seven?
• Suppose that either a member of the CS faculty or
  a student who is a CS major can be on a
  university committee. How many different choices
  are there if there are 37 CS faculty and 83 CS
  majors ?

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More Examples

• How many different license plates are available
  if each plate contains a sequence of three
  letters followed by three digits?
• What is the number of different subsets of a
  finite set S ?

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Example Using Both Rules

• Each user on a computer system has a password,
  which is six to eight characters long where each
  character is an uppercase letter or a digit. Each
  password must contain at least one digit. How
  many possible passwords are there?

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IP Address Example(Internet Protocol vers. 4)

• Main computer addresses are in one of 3 types
• Class A address contains a 7-bit netid ? 17,
  and a 24-bit hostid
• Class B address has a 14-bit netid and a 16-bit
  hostid.
• Class C address has 21-bit netid and an 8-bit
  hostid.
• Hostids that are all 0s or all 1s are not
  allowed.
• How many valid computer addresses are there?

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Example Using Both RulesIP address solution

• ( addrs) ( class A) ( class B) (
  class C)
• (by sum rule)
• class A ( valid netids)( valid hostids)
• (by product rule)
• ( valid class A netids) 27 - 1 127.
• ( valid class A hostids) 224 - 2 16,777,214.
• Continuing in this fashion we find the answer
  is 3,737,091,842 (3.7 billion IP addresses)

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Inclusion-Exclusion Principle(relates to the
sum rule)

• Suppose that k?m of the ways of doing task 1 also
  simultaneously accomplishes task 2. (And thus are
  also ways of doing task 2.)
• Then the number of ways to accomplish Do either
  task 1 or task 2 is m?n?k.
• Set theory If A and B are not disjoint, then
  A?BA?B?A?B.

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Example

• How many strings of length eight either start
  with a 1 bit or end with the two bit string 00?

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More Examples

• Hypothetical rules for passwords
• Passwords must be 2 characters long.
• Each password must be a letter a-z, a digit 0-9,
  or one of the 10 punctuation characters
  !_at_().
• Each password must contain at least 1 digit or
  punctuation character.

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Sol. Contd

• A legal password has a digit or puctuation
  character in position 1 or position 2.
• These cases overlap, so the principle applies.
• ( of passwords w. OK symbol in position 1)
  (1010)(101026)
• ( w. OK sym. in pos. 2) also 2046
• ( w. OK sym both places) 2020
• Answer 920920-400 1,440

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Tree Diagrams
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Pigeonhole Principle

• If k1 objects are assigned to k places, then at
  least 1 place must be assigned 2 objects.
• In terms of the assignment function
• If fA?B and AB1, then some element of B
  has 2 pre-images under f.
• i.e., f is not one-to-one.

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Example

• How many students must be in class to guarantee
  that at least two students receive the same score
  on the final exam, if the exam is graded on a
  scale from 0 to 100 points?

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Generalized Pigeonhole Principle

• If Nk1 objects are assigned to k places, then
  at least one place must be assigned at least
  ?N/k? objects.
• e.g., there are N280 students in this class.
  There are k52 weeks in the year.
• Therefore, there must be at least 1 week during
  which at least ?280/52? ?5.38?6 students in the
  class have a birthday.

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Proof of G.P.P.

• By contradiction. Suppose every place has lt
  ?N/k? objects, thus ?N/k?-1.
• Then the total number of objects is at most
• So, there are less than N objects, which
  contradicts our assumption of N objects! ?

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G.P.P. Example

• Given There are 280 students in the class.
  Without knowing anybodys birthday, what is the
  largest value of n for which we can prove that at
  least n students must have been born in the same
  month?
• Answer

?280/12? ?23.3? 24
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More Examples

• What is the minimum number of students required
  in a discrete math class to be sure that at least
  six will receive the same grade, if there are
  five possible grades, A, B, C, D, and F?

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Binomial Coefficients

• The number of r-combinations from a set with n
  elements is denoted by also called binomial
  coefficient.
• This number is the expansion of powers of
  binomial expressions such as (a b)n

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Binomial Coefficients

• THEOREM 1 The Binomial Theorem
• Let x,y,n be positive integer
• THEOREM 2 PASCALS IDENTITY
• Let n and k be positive integer with n gt k then
• THEOREM 3 Let n be positive integer, then
• More details in Section 5.4

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Examples

• (x y)4 ?
• What is the coefficent of x12y13 in the expansion
  of (x y)25?

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Permutations

• A permutation of a set S of objects is an ordered
  arrangement of the elements of S where each
  element appears only once
• e.g., 1 2 3, 2 1 3, 3 1 2
• An ordered arrangement of r distinct elements of
  S is called an r-permutation.
• The number of r-permutations of a set S with
  nS elements is
• P(n,r) n(n-1)(n-r1) n!/(n-r)!

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Example

• How many ways are there to select a third-prize
  winner from 100 different people who have entered
  a contest?

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More Examples

• A terrorist has planted an armed nuclear bomb in
  your city, and it is your job to disable it by
  cutting wires to the trigger device.
• There are 10 wires to the device.
• If you cut exactly the right three wires, in
  exactly the right order, you will disable the
  bomb, otherwise it will explode!
• If the wires all look the same, what are your
  chances of survival?

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More Examples

• How many permutations of the letters ABCDEFG
  contain the string ABC?

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Combinations

• The number of ways of choosing r elements from S
  (order does not matter).
• S1,2,3
• e.g., 1 2 , 1 3, 2 3
• The number of r-combinations C(n,r) of a set with
  nS elements is

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Combinations vs Permutations

• Essentially unordered permutations
• Note that C(n,r) C(n, n-r)

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Combination Example

• How many distinct 7-card hands can be drawn from
  a standard 52-card deck?
• The order of cards in a hand doesnt matter.
• Answer C(52,7) P(52,7)/P(7,7)
  52515049484746 / 7654321

52171074746 133,784,560
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More Examples

• How many ways are there to select a committee to
  develop a discrete mathematics course if the
  committee is to consist of 3 faculty members from
  the Math department and 4 from the CS department,
  if there are 9 faculty members from Math and 11
  from CS?

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Generalized Permutations and Combinations

• How to solve counting problems where elements may
  be used more than once?
• How to solve counting problems in which some
  elements are not distinguishable?
• How to solve problems involving counting the ways
  we to place distinguishable elements in
  distinguishable boxes?

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Permutations with Repetition

• The number of r-permutations of a set of n
  objects with repetition allowed is
• Example How many strings of length n can be
  formed from the English alphabet?

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Combinations with Repetition

• The number of r-combinations from a set with n
  elements when repetition of elements is allowed
  are C(nr-1,r)

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Combinations with Repetition

• Example How many ways are there to select 5
  bills from a cash box containing 1 bills, 2
  bills, 5 bills, 10 bills, 20 bills, 50 bills,
  and 100 bills? Assume that the order in which
  bills are chosen does not matter and there are at
  least 5 bills of each type.

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Combinations with Repetition

• Approach Place five markers in the compartments
• i.e., ways to arrange five
  stars and six bars ...
• Solution Select the positions of the 5 stars
  from 11 possible positions !

C(nr-1,5) C(75-1,5)C(11,5)
n7 r5
compartments and dividers
markers
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Combinations with Repetition

• Example How many ways are there to place 10
  non-distinguishable balls into 8 distinguishable
  bins?

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Permutations and Combinations with and without
Repetition
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Permutations with non-distinguishable objects

• The number of different permutations of n
  objects, where there are non-distinguishable
  objects of type 1, non-distinguishable
  objects of type 2, , and non-distinguishable
  objects of type k, is
• i.e., C(n, )C(n - , )C(n - -
  -- , )

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Permutations with non-distinguishable objects

• Example How many different strings can be made
  by reordering the letters of the word
• SUCCESS

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Distributing DistinguishableObjects into
Distinguishable Boxes

• The number of ways to distribute n
  distinguishable objects into k distinguishable
  boxes so that objects are placed into box i,
  i1,2,,k, equals

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Distributing DistinguishableObjects into
Distinguishable Boxes

• Example How many ways are there to distribute
  hands of 5 cards to each of 4 players from the
  standard deck of 52 cards?

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Generating Permutations and Combinations

• Often, we have to generate permutations or
  combinations to determine the solution
• A salesman must visit 6 different cities. In
  which order should these cities be visited to
  minimize total travel time?
• generate 6! 720 possibilities
• Find a subset of 6 integers that has their sum
  equals 100
• possible subset is 26 64 subsets

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Generating Permutations

• Any set with n elements can be place in
  one-to-one correspondence with the set 1, 2, 3,
  , n
• Generate permutation of the n smallest positive
  integers
• Replace these integers with corresponding elements

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Example

• procedure next permutation(a1a2an permutation
  of 1,2,,n not equal to n n-1 2 1)
• j n -1
• while aj gt aj1
• j j - 1 j is the largest subscript with aj lt
  aj1
• k n
• while aj gt ak
• k k - 1 ak is the smallest integer greater
  than aj to the right of aj
• interchange aj and ak
• r n
• s j 1
• while r gt s
• begin
• interchange ar and as
• r r 1
• s s 1
• end this puts the tail end of the permutation
  after the jth position in increasing order

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Generating Combinations

• How to generate all combinations of the elements
  of a finite set
• Combination is a subset
• Use the correspondence between subsets of a1,
  a2, , an

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Examples

• procedure next bit string(bn-1bn-2b1 bit string
  not eqal to 1111)
• i 0
• while bi 1
• begin
• bi 0
• i i 1
• end
• bi 1

See Section 5.6 for more details

• procedure next r-combination(a1, a2, , ar
  proper subset of 1, 2, ,n not equal to
• 
  n r 1, , n with a1 lt a2 lt lt ar)
• i r
• while ai n r i
• i i - 1
• ai ai 1
• for j i 1 to r
• aj ai j - i

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Prolog Tutorial

• From Learn Prolog Now
• http//www.coli.uni-saarland.de/kris/learn-prolog
  -now/lpnpage.php?pageidonline

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Prolog Basics

• Fact, Rules, and Queries
• 3 basic constructs in Prolog
• A collection of facts and rules is called a
  knowledge base (or a database).
• Prolog programming is all about writing knowledge
  bases.
• Prolog programs work by asking questions or
  querying about the information stored in the
  knowledge base.
• It makes sense for some applications such as
  Computational Linguistic and AI.

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Prolog Basics

• Fact, Rules, and Queries
• 3 basic constructs in Prolog
• A collection of facts and rules is called a
  knowledge base (or a database).
• Prolog programming is all about writing knowledge
  bases.
• Prolog programs work by asking questions or
  querying about the information stored in the
  knowledge base.
• It makes sense for some applications such as
  Computational Linguistic and AI.
• The best way of learning Prolog is by learning
  Knowledge Base (KB)

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Prolog Basics
woman(mia).woman(jody).woman(yolanda).playsAirG
uitar(jody).

• KB1 facts
• Each statement in Prolog ends with .
• Prior to infer or conclude anything, we have to
  consult KB first.
• To consult KB is to load it to Prolog
  workspace.

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Prolog Basics
listensToMusic(mia). happy(yolanda). playsAirGuita
r(mia) - listensToMusic(mia). playsAirGuitar(yol
anda) - listensToMusic(yolanda). listensToMusic(y
olanda)- happy(yolanda).

• KB2 rules
• There are no fact about mia plays air guitar in
  the KB
• But there is a rule about mia plays air guitar
• Prolog use Modus Ponens to infer that mia plays
  air guitar is true
• head - body read as if the body is true, then
  the head is true

rules
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Prolog Basics
happy(vincent). listensToMusic(butch). playsAirGui
tar(vincent)- listensToMusic(vincent),
happy(vincent). playsAirGuitar(butch)-
happy(butch). playsAirGuitar(butch)-
listensToMusic(butch).

• KB3 more rules
• 5 clauses 3 rules and 2 facts
• 3 predicates happy, listenToMusic,
  and playsAirGuitar
• In a KB, there is no fact saying
  vincent
  listens to music
• Hence, the query playsAirGuitar(vincent) is false
• head - body1, body2 ? (body1 ? body2) ? head

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Prolog Basics
woman(mia). woman(jody). woman(yolanda).
loves(vincent,mia). loves(marcellus,mia). loves(p
umpkin,honey_bunny). loves(honey_bunny,pumpkin).

• KB4 variables
• A variable is in CAPITAL
• Uses a variable to ask
  whose
  property it is.
• means or
• , means and
• loves(marcellus,X), woman(X)
  should read whom is
  a woman
  Marcellus love?
  
  ( whom does Marcellus love and is woman?)

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Prolog Basics
loves(vincent,mia). loves(marcellus,mia). loves(pu
mpkin,honey_bunny). loves(honey_bunny,pumpkin).
jealous(X,Y) - loves(X,Z),loves(Y,Z).

• KB5 more variables
• Now, we use variable in rules.
• Predicate jealous(X,Y) means that an
  
  individual X will be jealous of an
  
  individual Y if there is some
  
  individual Z that X
  loves, and Y loves
  
  that same individual Z too.
• This is more general than previous KB.
• jealous(marcellus,W) asks can you find
  
  an individual W such that
  Marcellus is jealous of W?
• Furthermore, suppose we wanted Prolog to tell us
  
  about all the jealous people what query would we
  pose?

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Prolog Basics

• Terms
• All facts, rules, and queries are made of terms.
• There are 4 kinds of term in Prolog atoms,
  numbers, variables, and complex terms (or
  structures).
• Atoms and numbers are lumped together under the
  heading constants, and constants and variables
  together make up the simple terms of Prolog.
• Please consult websites, books or other resources
  for further details.

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Python Tutorial

• From Learn Python in 10 minutes
• http//www.poromenos.org/tutorials/python

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Python Basics

• Properties
• dynamically, implicitly typed (i.e., do not have
  to declare the type )
• case sensitive
• object-oriented
• Use interpreter
• To get help in Python, type help(object) at the
  prompt.
• Type dir(object) to ask Python how to use that
  method

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Python Basics
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Python Basics

• Syntax
• No mandatory statement terminate character
• Block is specified as indentation
• Statement that expects indentation ends in colon
  ()
• Comments start with pound () and are single line
• Use multi-line strings for multi-line comments
• Values are assigned with the equal sign ()
• is used for equality testing
• and - are used for increment and
  decrement, respectively

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Python Basics
77
Python Basics

• Data types
• Lists one-dimensional array
• Tuples immutable one-dimensional array
• Dictionaries associative array (hash table)
• Index starts from 0, negative number counts from
  the end toward the beginning of list, -1 is the
  last item
• Variables can point to function
• Can access array ranges using colon ()

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Python Basics
79
Python Basics

• String
• Use either single quote or double quote marks (
  or ) to create string
• Multi-line strings are enclosed with triple
  double or single quote
• To fill a string with values, use s and a tuple

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Python Basics

• Flow Control Statements
• They are while, if, and for. No select.
• Use for to enumerate through a list
• Use range(number) to obtain a list of numbers

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Python Basics
82
Python Basics

• Function
• Declares with def keyword
• Optional arguments are set in the function
  declaration after mandatory arguments by being
  assigned a default value.
• For named arguments, the name of the argument is
  assigned a value.
• Functions can return a tuple.
• Lambda functions are ad hoc functions that are
  comprised of a single statement.
• Parameters are passed by reference, but immutable
  types (tuples, ints, strings, etc) cannot be
  changed. This is because only the memory location
  of the item is passed, and binding another object
  to a variable discards the old one, so immutable
  types are replaced.

83
Python Basics
84
Python Basics

• Importing
• Use import libname keyword to import external
  library
• Or use from libname use funcname to import
  specified function from external library

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Python Basics

• Miscellaneous
• Conditions can be chained. 1 lt a lt 3 checks that
  a is both less than 3 and more than 1.
• You can use del to delete variables or items in
  arrays.
• List comprehensions provide a powerful way to
  create and manipulate lists. They consist of an
  expression followed by a for clause followed by
  zero or more if or for clauses.
• Global variables are declared outside of
  functions and can be read without any special
  declarations, but if you want to write to them
  you must declare them at the beginning of the
  function with the "global" keyword, otherwise
  Python will bind that object to a new local
  variable (be careful of that, it's a small catch
  that can get you if you don't know it).

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Python Basics
See these links for further tutorials http//docs.
python.org/tutorial/ http//www.sthurlow.com/pytho
n/
87
Homework

• Section 5.1
• 2, 10, 18, 25, 33, 34, 43
• Section 5.2
• 1, 8, 10, 15, 32
• Section 5.3
• 8, 14, 24, 33, 35, 39
• Section 5.4
• 15, 28, 32
• Section 5.5
• 1, 13, 15, 16, 30, 37, 50, 59
• Section 5.6
• 10, 13
• Supplementary
• 4, 11, 22, 27, 36, 42