Discrete Math Lecture 5: Counting

461191 Discrete Math Lecture 5: Counting
San Ratanasanya CS, KMUTNB

Today’s topics Review of Induction and Recursion Administrivia
Basics of Counting Prolog Tutorial Python Tutorial

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(k+1)) 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(k+1) is true in the inductive step instead of just showing only P(k)  P(k+1) as in mathematical induction

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 k+1 horse also have the same color. Second set of k horse having the same color k+1 horse having the same color First set of k horse having the same color

Example “For any 2n x 2n board, we can chase Bill to the corner.”
Proof: Basis Step: no board (n=0), 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 2n+1 x 2n+1 We are done!

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

Example Give a recursive definition of

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

Example procedure multiply (m, n: integers) if n < 0 then a := -n
else a := n k := 0 x := 0 while k < a begin x := x + m k := k + 1 end S1 S3 S2 if n < 0 then product := -x else product := x S4 Prove that this program is correct. Let p: “m and n are integers” and q: p  (a=|n|). Then we can prove that p{S1}q is true. Let r: q  (k=0)  (x=0). Then it can be proved that q{S2}r 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: “x=mk  k  a” or “x=ma  a=|n|”. 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 r{S3}s is true. Now that we prove S1, S2, S3, and S4 are true so, by composition rule we could show that [p{S1}q  q{S2}r  r{S3}s  t{S4}u]  p{S1;S2;S3;S4}u is true. p{S}u is correct.

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

Basics of Counting

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

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

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 ...

Discrete Mathematics and its Applications
4/21/2017 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? (c) , Michael P. Frank

4/21/2017 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B|=|A|+|B|” (c) , Michael P. Frank

4/21/2017 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 ... (c) , Michael P. Frank

4/21/2017 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? (c) , Michael P. Frank

4/21/2017 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B|=|A|·|B| (c) , Michael P. Frank

4/21/2017 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 ? (c) , Michael P. Frank

4/21/2017 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 ? (c) , Michael P. Frank

Example Using Both Rules
Discrete Mathematics and its Applications 4/21/2017 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? (c) , Michael P. Frank

IP Address Example (Internet Protocol vers. 4)
Discrete Mathematics and its Applications 4/21/2017 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? (c) , Michael P. Frank

Example Using Both Rules: IP address solution
Discrete Mathematics and its Applications 4/21/2017 Example Using Both Rules: IP 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) (c) , Michael P. Frank

Inclusion-Exclusion Principle (relates to the “sum rule”)
Discrete Mathematics and its Applications 4/21/2017 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B|=|A||B||AB|. (c) , Michael P. Frank

4/21/2017 Example How many strings of length eight either start with a 1 bit or end with the two bit string 00? (c) , Michael P. Frank

4/21/2017 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 Each password must contain at least 1 digit or punctuation character. (c) , Michael P. Frank

4/21/2017 Sol. Cont’d 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) = (10+10)·( ) (# w. OK sym. in pos. #2): also 20·46 (# w. OK sym both places): 20·20 Answer: −400 = 1,440 (c) , Michael P. Frank

Tree Diagrams

4/21/2017 Pigeonhole Principle If k+1 objects are assigned to k places, then at least 1 place must be assigned ≥2 objects. In terms of the assignment function: If f:A→B and |A|≥|B|+1, then some element of B has ≥2 pre-images under f. i.e., f is not one-to-one. (c) , Michael P. Frank

4/21/2017 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? (c) , Michael P. Frank

Generalized Pigeonhole Principle
Discrete Mathematics and its Applications 4/21/2017 Generalized Pigeonhole Principle If N≥k+1 objects are assigned to k places, then at least one place must be assigned at least N/k objects. e.g., there are N=280 students in this class. There are k=52 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. (c) , Michael P. Frank

4/21/2017 Proof of G.P.P. By contradiction. Suppose every place has < 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! □ (c) , Michael P. Frank

4/21/2017 G.P.P. Example Given: There are 280 students in the class. Without knowing anybody’s 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 (c) , Michael P. Frank

4/21/2017 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? (c) , Michael P. Frank

Binomial Coefficients
Discrete Mathematics and its Applications 4/21/2017 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 (c) , Michael P. Frank

Binomial Coefficients
Discrete Mathematics and its Applications 4/21/2017 Binomial Coefficients THEOREM 1: The Binomial Theorem Let x,y,n be positive integer THEOREM 2: PASCAL’S IDENTITY Let n and k be positive integer with n > k then THEOREM 3: Let n be positive integer, then More details in Section 5.4 (c) , Michael P. Frank

4/21/2017 Examples (x + y)4 = ? What is the coefficent of x12y13 in the expansion of (x + y)25? (c) , Michael P. Frank

4/21/2017 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, , An ordered arrangement of r distinct elements of S is called an r-permutation. The number of r-permutations of a set S with n=|S| elements is P(n,r) = n(n−1)…(n−r+1) = n!/(n−r)! (c) , Michael P. Frank

4/21/2017 Example How many ways are there to select a third-prize winner from 100 different people who have entered a contest? (c) , Michael P. Frank

4/21/2017 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? (c) , Michael P. Frank

4/21/2017 More Examples How many permutations of the letters ABCDEFG contain the string ABC? (c) , Michael P. Frank

4/21/2017 Combinations The number of ways of choosing r elements from S (order does not matter). S={1,2,3} e.g., 1 2 , , The number of r-combinations C(n,r) of a set with n=|S| elements is (c) , Michael P. Frank

Combinations vs Permutations
Discrete Mathematics and its Applications 4/21/2017 Combinations vs Permutations Essentially unordered permutations … Note that C(n,r) = C(n, n−r) (c) , Michael P. Frank

4/21/2017 Combination Example How many distinct 7-card hands can be drawn from a standard 52-card deck? The order of cards in a hand doesn’t matter. Answer C(52,7) = P(52,7)/P(7,7) = 52·51·50·49·48·47·46 / 7·6·5·4·3·2·1 17 10 7 2 8 52·17·10·7·47·46 = 133,784,560 (c) , Michael P. Frank

4/21/2017 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? (c) , Michael P. Frank

Generalized Permutations and Combinations
Discrete Mathematics and its Applications 4/21/2017 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? (c) , Michael P. Frank

Permutations with Repetition
Discrete Mathematics and its Applications 4/21/2017 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? (c) , Michael P. Frank

Combinations with Repetition
Discrete Mathematics and its Applications 4/21/2017 Combinations with Repetition The number of r-combinations from a set with n elements when repetition of elements is allowed are C(n+r-1,r) (c) , Michael P. Frank

Discrete Mathematics and its Applications 4/21/2017 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. (c) , Michael P. Frank

Discrete Mathematics and its Applications 4/21/2017 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(n+r-1,5)= C(7+5-1,5)=C(11,5) n=7 r=5 compartments and dividers markers (c) , Michael P. Frank

Discrete Mathematics and its Applications 4/21/2017 Combinations with Repetition Example: How many ways are there to place 10 non-distinguishable balls into 8 distinguishable bins? (c) , Michael P. Frank

Permutations and Combinations with and without Repetition
Discrete Mathematics and its Applications 4/21/2017 Permutations and Combinations with and without Repetition (c) , Michael P. Frank

Permutations with non-distinguishable objects
Discrete Mathematics and its Applications 4/21/2017 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 … , ) (c) , Michael P. Frank

Permutations with non-distinguishable objects
Discrete Mathematics and its Applications 4/21/2017 Permutations with non-distinguishable objects Example: How many different strings can be made by reordering the letters of the word SUCCESS (c) , Michael P. Frank

Distributing Distinguishable Objects into Distinguishable Boxes
Discrete Mathematics and its Applications 4/21/2017 Distributing Distinguishable Objects into Distinguishable Boxes The number of ways to distribute n distinguishable objects into k distinguishable boxes so that objects are placed into box i, i=1,2,…,k, equals (c) , Michael P. Frank

Distributing Distinguishable Objects into Distinguishable Boxes
Discrete Mathematics and its Applications 4/21/2017 Distributing Distinguishable Objects 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? (c) , Michael P. Frank

Generating Permutations and Combinations
Discrete Mathematics and its Applications 4/21/2017 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 (c) , Michael P. Frank

Generating Permutations
Discrete Mathematics and its Applications 4/21/2017 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 (c) , Michael P. Frank

4/21/2017 Example procedure next permutation(a1a2…an: permutation of {1,2,…,n} not equal to n n-1 …2 1) j := n -1 while aj > aj+1 j := j - 1 {j is the largest subscript with aj < aj+1} k:= n while aj > 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 > 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} (c) , Michael P. Frank

Generating Combinations
Discrete Mathematics and its Applications 4/21/2017 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} (c) , Michael P. Frank

4/21/2017 Examples procedure next bit string(bn-1bn-2…b1: bit string not eqal to 11…11) 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 < a2 < … < ar) i := r while ai = n – r + i i := i - 1 ai := ai + 1 for j := i + 1 to r aj := ai + j - i (c) , Michael P. Frank

From “Learn Prolog Now”
Prolog Tutorial From “Learn Prolog Now”

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.

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)

Prolog Basics KB1: facts Each statement in Prolog ends with ‘.’
woman(mia). woman(jody). woman(yolanda). playsAirGuitar(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.

Prolog Basics KB2: rules
listensToMusic(mia). happy(yolanda). playsAirGuitar(mia) :- listensToMusic(mia). playsAirGuitar(yolanda) :- listensToMusic(yolanda). listensToMusic(yolanda):- 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

Prolog Basics KB3: more rules 5 clauses: 3 rules and 2 facts
happy(vincent). listensToMusic(butch). playsAirGuitar(vincent):- listensToMusic(vincent), happy(vincent). playsAirGuitar(butch):- happy(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

Prolog Basics KB4: variables A variable is in CAPITAL
woman(mia). woman(jody). woman(yolanda). loves(vincent,mia). loves(marcellus,mia). loves(pumpkin,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?”)

Prolog Basics KB5: more variables Now, we use variable in rules.
loves(vincent,mia). loves(marcellus,mia). loves(pumpkin,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?

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.

From “Learn Python in 10 minutes”
Python Tutorial From “Learn Python in 10 minutes”

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

Python Basics

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

Python Basics

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 (:)

Python Basics

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

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

Python Basics

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.

Python Basics

Python Basics Importing
Use import [libname] keyword to import external library Or use from [libname] use [funcname] to import specified function from external library

Python Basics Miscellaneous
Conditions can be chained. 1 < a < 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).

Python Basics See these links for further tutorials

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

Discrete Math Lecture 5: Counting

Similar presentations

Presentation on theme: "Discrete Math Lecture 5: Counting"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Discrete Math Lecture 5: Counting

Similar presentations

Presentation on theme: "Discrete Math Lecture 5: Counting"— Presentation transcript:

Similar presentations

About project

Feedback