Discrete Mathematics For Computer Scientists And Mathematicians Discrete Mathematics and Theoretical Computer Science, 4 conf., DMTCS Discrete Mathematics for Computer Scientists and Mathematicians book. Read 9 The pdf of this book can be found on teshimaryokan.info Using the Strong Form of Mathematical Induction Using Discrete Mathematics in Computer Science CHAPTER 2 .. changes we make will be posted at teshimaryokan.info

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Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's as mathematical foundation of computer science, this book present a artificial intelligence. 𝗣𝗗𝗙 | These are notes on discrete mathematics for computer scientists. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of. Mathematical Induction. The Principle of Mathematical Induction. .. In many computer science departments, discrete mathematics is one of the.

The Sloan Foundationhas played a Florida State University of the need for discrete major role in educating the academiccommunity mathematics 'uq. The ordered pair of a and 6. False k Begs the m Begs n Begs the question. N is not divisible by 3 are than n integers. The laws of algebra may apply but there arerestraintson their for instance.

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If the inferenceis faulty. AB and lines the then Hence. If a majority of the Cabinetare in will call a special The Senate can reacha compromise. AB know We spherical. BC are equal. Mackis not guilty. The governor will call a special session only if the Senate cannot the governor reach a compromise. Either Mack Mike is not is not guilty or truth. David will pass the course. If a majority of the Cabinet are in agreement.

If diamonds then gold is sellingcheaply. Lowellis not studying if the The governor will call a specialsession Senate cannot only reach a compromise. Goldis not sellingcheaply. Theearthis 6. If not the moonis spherical. The then agreement. C is on the perpendicular bisector is equidistantfrom A and B.. If lines AB the then alternate are parallel. If the graphsareisomorphic. If a majority of the Cabinet are in will call a specialsession.

If two-colorable. Theirdegree different Fill IV.. Ifitis not sun will out. If the graphs are isomorphic. The governor will the then agreement.. If Joedoes graduate.

But angles Hence. The new car.. The Senate reach a governor a compromise. If Melindais late. The graph G is not two-colorable.. The Senate the governor a compromise. C does not if The graphs have numbers different of edges.. If wages the cup it is then water. If Tallahassee is not in Florida.

Determine patterns a combination of indicate or invalid. Golf balls arenotsoldin Chicago. Foundations is valid whether each of the following inference V. Joe can carry it. If the cup is lighterthan Hence. If the cup is styrofoam. If the inferencepattern is invalid.

If thereisa depression. If the inference truth values which will produce a counterexample. Tallahassee Hence. If the similar. Fallacy 5. If Hence. Selected for Section 1.

Joe mathematician. Verify symbols 59 Inferences Logical assistant. Rule 1. EitherClifton French.. The b triangle c Henceif II. If Joeisan early riser. Hence the graphs 8. If Joeisambitious.

No Hence. If Clifton in France. Valid 1 -rVu. Thenthe premises true 5. The methods nine referred frequently known methods of of proof and in two basic inference patterns methods for proving implications. Every conclusion must be made according to valid inference We must skeleton emphasize outline patterns. Foundations Let the statements have of one are truth-values. In developinga proof the of student should follow the skeleton outline of some indicatedmethod to to the more will have add flesh bones than student the but.

Vacuous proof of implicationp q true 3. Trivial then. If it is possible to establishthat q is true.. This method contradiction. Proof ofp that leads us to and therefore the then q proof by a Assumep A -g is true. Indirect implication p Consequently. Section Brief 1. V If p cases. V r by proof by contradiction. If for someotherreasonswe conclude we must verify that p is. V is nothing proof [ p V q finite V p2 of method syllogism: Conditional proof. If X2. The p to a contradiction.

We might observe that this methodof elimination of cases is quite similarin form however. Section of Proof Methods 1. Example is a number suchthat a2. First we list someelementary of direct Examples direct of examples proofs. Now since we are discussing numbers. Example p represent someinteger.

We may. Since by. Proof of equivalence. Let us recallthat integer. Toprove if a proposition q is true frequently p is true if and proof into two of course. Example then Supposethat 1 is is suchthat a - 2 is divisible by 3. A direct proof of the statement proceeds as follows: Having proof. Using Assume of algebra. The statement p that the remainders rx and r2 are equal. Two integers a and b have 1. The integersq and r are called. But attempt to prove that a divisionalgorithm.

Foundations since Therefore. If a and b are integers where b is positive. But then. Of a is course. Exampleofa proof of only if r given by remainder the the equivalence. Example divided by the positive integer n iff the integer a same the remainder when by n.

The smallest divisor prime Input: The smallest prime divisor of n. We list of all primes lessthan or equal to the square root of n is available. The above algorithm Algorithm 1. By DeMorgan's are we familiar b is a and integers even. If the productof two eithera is to illustrate now wish first example Our contrapositive. If n is an integer. Example contrapositive a and integers or assume a smallerfactor of n.

Example and b 4- 8. Section of Proof of an Methods 1. Example birthdays this people proposition and no pair month. Show that the sum of someset numbers is at least Examples of proofs contradiction.

Go to Step 3. If n is even. But then sinceeach by assumingthat theroom by contradiction of people have their in the birthdays in some is born person their same month. If For instance. Foundations Method: P be the set of all primes lessthan Let or equal to s. Consider the integer n.

Find the largest integer s less than or equal to the squarerootofn. This conclusion is in violation of the well-known fact that there are 12 months. If no x G P is the currently considered of n. If mx 4. The same 1. X10 is just the sum of the first can observe is equal to X4 the number respectively. X4iX5 Example pigeonhole or the. X2 fact.

Let mum2i. X2 which we becomes is a this Example white. We are to prove Let to mean the smallest to mean the largest integer this second the terminology. We do this by contradiction. In since mx A 4. This clearly equals the total number is a contradiction of the floor of A A L9. If n is the number of in of ith number the then we hole. The proof that some m. Then some row contains at least 9 at least 5 students. Supposethat a patient is given a prescription of 45 pills with the instructions to take at least one pill per day for 30 days.

Gi Suppose one chair. Section Methods 1. If impliesa 4. The the average number most because average number per columnis 4. Tosee taken let a. If letters delivered receivedat most8 letters. Then prove that which the patient takes there must be a periodofconsecutive days during of studentsperrow is 8. In any group of people there must be at leastonepair with the 1. This choice for pigeonholes would have the feature that if 2 integers a and b are in the same hole.

This latter approach..

The division proved algorithm impliesthat the set of integers can be partitioned into 7 ax Say a and b are in the same pigeonhole. The class is useful in the next concept example.

We there are at least 2 pigeons in somehole or in this conclude that which are tliere are at least 2 of the remainders framework.. If a and b are in [0] or [5]. We use the following ploy: By must have the same the definition of congruenceclass. If we apply what we learned in 8 above. If the integera is such that a. If the. If x value of x. If p isan odd prime. Or in [8]. We have this in example already proved 1.

This fact will be proved integers. By conditional proof i canbechanged the product of two integers be. Use the division casesto a prove Every odd and algorithm the by cases proof or elimination of following: Sincea is primeand a does not weapon. Suppose to prove: But then if we multiply this equation by c. If a is a prime integerand a divides b or a divides c. The square of any integer is eitherofthe form of the form 9n. Example that proving: We iii yet. Foundations i If a is a primeinteger.

Prove or disprove: If a divides b and c dividesd. If a is an odd integer. If the sum of 11 real numbers than Prove for an integer a. If x is an integer. If a b is is evenand c odd. Prove for integers a and 6: For integers a. If the product of a by contradiction: Prove that he must have hiked at least 12 miles within a certain period of two consecutive hours. Prove integer 8. Foundations 7. Suppose that the circumferenceof a circular wheel is divided into 50 sectorsand that the numbers 1 and 50 are randomly assigned to c Given them 37 integers.

Suppose that hour and 4 miles the twelfth hour and hiked a total of 71 milesin 12 hours. Use the pigeonhole a rationalnumber to show principle must.

If n is a positive A positive n. Show was used for at least 17hours. Use square rootof2 isnot a rational implies x is even. Show that it is possible to align the two disks so that or more of the sectionson the inner disk have their colors matched with the sections on the outer disk.

Give Mn The circumference of two sections each. Forthe outer and sections concentric disk. Section that for an A typewriter is used for hours over a period of that on some pair of consecutivedays. Forthe inner disk are painted red or white in an arbitrary manner. A argument a contrapositive is not integer perfect 5x an integer. Give integer a prime.

Then the apply principle. Find no other pair exists. If are there pairs of different were chosen? From past will no more knowsthat she she than 60 hours require experience of study. Show She alsowishes at least 1 hour per day. Thus5 a primes and 7 are twin primes.. Provethat in a group of n people there are two who have the same number of acquaintancesin the group. Find a set of triple primesand prove a that Prove Write b Twin are primes a and b such that b.

Given the information that no human beinghas morethan What is the largest with the same number of hairson their integer that canbeused for n in the following assertion? There are in Florida n with the same number of hairs on their persons A student heads.

Foundations at least tee that a 10 one of books language b 6 French. Find another Thus. Find 4 other pairs of twin primes. Let dti be the distance from P. Give a direct proof that any is a primitive triple iff a2 Pythagoreantriple triple. Demonstratethat for at least one pair of the points P. Let S be a of square side each hole is to be in order contents of at least one in reverse form.

Provethat if a and b are in A.. Showthat there is some decimal digits are Pythagorean A be the set of all integers that can bewritten Let as a sum of two squares of integers. Choose any five points the interior of S. Given distributed among k pigeonholes: Suppose that the pigeons are distributedso that How many pigeonsmust we choose nonempty.

Let p be a of power for If the of 10 nonnegative somesetof of these integers of 5 of them is at least If a is an k is a divisor integer. Euclidean the for finding algorithm the greatest common divisor. That is. In calendar any there be. Then if divisors integers D a D D b isthe setofall positive common to both a and b. What is the smallest number Let Xi 1. This is the basis for 6.

Let's hold the outerdisk fixed the possible alignments. We actually prove a stronger result. Let x. By this something of what makesa valid argument.

Suppose which implies From a mathematicalpoint of view. Putting it very roughly. Joeis We sucha treatment. Foundations are let are not susceptible to the following argument: First-order logicis that part of logicwhich and G represents In case as the of content the emphasizes the sentences involved form arguments.

Gauss is a sentence becomesa proposition. An open variables from a set Uis a function or proposition f: Likewise determine whether or not that unspecifiedpersonis rational? If we the sentence then. In all cases. All isosceles are equiangular. Mount Everest. Foundations Some 1. G 4 is false. Heis a Everest. There are some real numbers that are not rational numbers. R is true.

Certain declarative sentencesinvolve indicate words that quantity such as all. Consider the following statements: Some parallelogramsare squares.

R V2 is false. For all x. Each Not with flirting is one and only 8. For every x. Not are All smokers 7. For can be write of discourse we can prime integer. It we and represents shall each of phrases: There For each x. Not all prime integersareodd.

Section Order Logic First 1. F x proposition open write can we mathematician least onex suchthat. It by the variable of each the since all have represents following phrases. There is an x suchthat.. P P is defined as the subset for a given set the truth of open proposition of the universe of x such that all P x is true. The universal quantifier often followed by an implication becausea universal is most statement often of the form it also has any x.

The universe a role when plays significant analyzing sentences. In translating sentences with into we find a quantifiers symbols is very common. Likewise if the sentence universe but the isfalse if the the number includes. Let quantified eight and their abbreviatedmeaningin statements the list: F x ] at least one none true Vx.

Of modifier the using course. F x Meaning meaning. F c a glorified version of than. Revisited Laws DeMorgan's Let us recordthe following observations: Foundations information about the negationof this the first statement we have Vx.

F x For instance. In the light of thesecomments. F x and. F c is true. Proof sufficient x all all F x. F c is false. The Is True Statement Vx. F x 9Vx. The following chart shows when eachmain type of quantified proposition is true and when it is false.

Proof of assertions existence proofs nonconstructive. Proof by exhaustion. This one assertion Vx. Proof 2. To show 3x9 F x is true. F x in that c the universesuch example to the c is called a counterexample fact.

A of the statement form Vx. This type is the only situation where an exampleproves anything. False for all c. Foundations rather is true or sometimes. But the proof is nonconstructive. Such a a value c involves a proof by commonly c. F x such that F c is true. Letn be a positive and define of partitions of n. We in in combinations. Section First Order 1.

These combinationsbecomeparticularly important the For the case of sentences involving one variable. Foundations In is any predicate involving if P x.

The first first is true. Graphical representationof two What this diagram Vy P x. Vy Vx Vx Vy 3y Vx two with sentence. Let us restate all the. The second x and y is 5. Vx P list: One sentences two real numbers: Section First Order Logicand 1.

All the can birds following and variables. Let us with only applying the rulefor illustrate. Not all birds canfly. Then expresseachofthe following x x is even or is a a perfectsquare.

If x is a is and not even. Foundations k x is an odd 1 For all m For each integer and x is prime. Determine or falsity of the following the truth universe U is the setof integers. Let x is xy integer odd and x is prime. There is an integerx such that x is even and x is prime. For each h Vx. P 2 Lagrange P x is a - x2 and Foundations Jackson is a baseballplayer. If xl9 x X -Xand i-l i-l Can generalize? Note that 4 5 Xi a string that observe Thus table: Derive can be degree of P X.

Indicate how to e Indicate 3 f such all 0 through Neithercan 9 be placed in any corner position. Such an arrangement is calleda magicsquare. The squares so that location there is a limited number of observe that 5 must occupy the positions observe second row and second column position.

First discover what each row sum and thereforewhat each a column and diagonal sum must be. Suppose that we have Problem Squares Magic nine lxl tilescontaining a 3 x 3 square and problemis to the numbers across sumof the place each row. Note that any one of these can be digits through transformed into other by either a rotation or a any any legitimate for for 9. Define a as entries.

Next 7. The square. SelectedAnswers for Section1. P x will from Universal an assertions prove of these treatment some of illustrate of inference is assumed to careful A when the deleted Fundamental Rule 5.

The universal and existential assertion. The negation planar. Propositions 97 squares. P s for all c'. This on the proper if we use of this rule. Pic is true quantifier may c of Pic for. Pix and 3x. Rule there c in the that if a statement is true of object by assigningit a name. Qix areboth then we can conclude Pic A true.

This rule permits the the objects the universal quantification of assertions. The next rule. Pic for some c ' Notethat the element c is not arbitrary as it was in Rule 5. Qix are both A Qic is false for every but Pic c in the universe of integers.

Universal element each for is an Specification. Foundations be Pic is true for to obtain dropped This rulemay be an object arbitrary universe. Pix is the universesuchthat takes the form 3x. This rule holds universe. Fundamental Rule 6. Pic Thus. In symbols. An quantifiers is the of because beyond are involved. Reasons Assertion 1. Example a king. Section c in element some 8. All are men All kings fallible. P x inference.

Existential the universe. It is b Somenegative following: Premise 2 3xtL x L a Exercises Premise 1 5. Rule Rule 1 8 1. Example the following argumentand checkfor Symbolize validity: Lions animals..

A formal is as proof follows: No rational to the roots equation X2. All integers are rational numbers. Use following propositions involve predicates these sets. Some animals Therefore. Someintegersare of powers 2. The a cigarettes are hazardous All Smokums are cigarettes. Some rational g Therefore. Prove or disprovethe validity arguments: David's b hearts. Use the properties to concluderelationshipsbetween Venn diagrams to check the validity of the arguments.

All of 3. All All some scientists Some astronauts Hence. The proofof 1 Vjc. David's dog Thenthe inferencepatternis: Some females are not mothers. All fathers i graduates. Some fathers. Foundations Therefore. Let us describe of Mathematical Induction. P k implies P k -f 1. Thereisa prooftechnique that such conjectures. As we certain statements as accepting other statements on the basisof The inductive aspect. Section 6 A a 7 H a Specification 4.

The Principle is true positive now. Let P n be integer n. Let us use this approach of Example problem for the sum of the first n positive integers. Foundations methodof principle of mathematicalinductionisa reasonable 1 tells us that P l is true. This is like induction stand domino conclusion n. Continuing 2 implies we would ultimately reach the conclusion is true for any fixed that P n Now the proof for part ofmathematical positiveintegern.

The principle the game we played as childrenwhere we would if over fell one it would collide with Then part 2 ofthe principle. Let P n us use mathematicalinductionto prove statement: Then 2 and the fact using that part 1 tellsus that P l is true. Section Mathematical Induction 1. Still we may not see the pattern at first. Inductive Step. Any nonempty integers. This be considered and the formula holds for k k. Inductive Hypothesis.

To do this. Assume the statementP n 2. Then we showthat. But Basis of Induction. P k al9 a2. N is not divisible by 3 are than n integers. Example prime integers. In case b. Example and 9-cent 1. Therearetwo 5. Prove that it is possibleto makeup any n-cents using only 5. Let P n bethe Suppose the Postal Departmentprints stamps.

The idea of a subroutine which is common. Prove that 1. Such a recursive subroutine. Informally the name of a use the techniqueof defining itself where it is be a set. We now prove this. Recursion In programming computer execution of a procedure is usually the evaluation of a function or the achieved at machine-languagelevel by subroutine.

Since greater 1. Foundations Example Let P n statement: In other words increases. P n Thus. We we 4 1 is true on the basisof our assumption. LetF bea given function set into S. Section Mathematical 1.

Let s0 be a fixed element of of S. A thorough discussion of recursive definitions and all the machinery that certain functions are well-defined to prove recursive necessary by is beyond our intentions for this book. This stateof affairs makes the recursive definition vulnerable to the chargeof circularity.

N nonnegative unique integers. We shall be content to definitions mention only the following theorem. G n is just.

The condition 1 of the Recursion Theorem condition and condition 2 Both rule. Another of this [16] or written N. Buck [7]. Strong MathematicalInduction: Let which..

To prove properties of sequences form of the principle of another induction. Chapter likethe Fibonacci This mathematical principleof mathematical induction. But in we using are strong. Strong InductiveHypothesis.. F0-l-Fl 2. Inductive Showthat P k 1 is Step. Show P q are all true..

Then P n is true 1 such P 2. We satisfying two initial have we example the uniqueness 5. Basis of Induction.. A relation recurrence and discuss this of form stronger existence conditions which are required defined in terms of both n and the Recursion Theorem is neededto prove the of a function these conditions.

P q areall true. Section The famous Fibonaccisequenceis defined l. Foundations we assume not only P k but also P k. By step. To track Flights around your favorate airports, click here. Gallier Publisher: English ASIN: Book Description This book gives an introduction to discrete mathematics for beginning undergraduates. We also discuss matchings, covering, bipartite graphs. About the Authors Jean H. Gallier is a researcher in computational logic at the University of Pennsylvania, where he holds appointments in the Computer and Information Science Department and the Department of Mathematics.

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