Section 4.3. Boolean functions
|
|
- Lucas Stephens
- 5 years ago
- Views:
Transcription
1 Sectio 4.3. Boolea fuctios Let us take aother look at the simplest o-trivial Boolea algebra, ({0}), the power-set algebra based o a oe-elemet set, chose here as {0}. This has two elemets, the empty set, which is (bottom), ad the set {0}, which is. Let us write 2 for this algebra; thus, 2 = {, }, with the total orderig give by <. t is obvious, either because we are havig a two-elemet total orderig, or because we are havig a algebra of sets, that the Boolea operatios are as follows: = = = = = = = = - = - =. We read as true, as false ; we read the operatio as ad, as or, - as ot. this way, the two-elemet Boolea algebra becomes a algebra of truth-values, ad becomes the basis of propositioal logic. propositioal logic, we aalyze seteces ito costituet parts out of which the setece is built up usig the coectives: (cojuctio; "ad"), (disjuctio; "or"), (egatio; "ot"; the differece to the "mius" sig, -, is iessetial), ad two more: (coditioal; "if..., the...") ad (bicoditioal; "if ad oly if"). We also call a setece of the form A B a cojuctio, its terms A ad B the cojucts i the setece. As idicated i the previous paragraph, the operatio of cojuctio is the oe that forms A B out of A, B. A B is a disjuctio; A ad B are its disjucts. A B is a coditioal; A is its atecedet, B its succedet. A B is a bicoditioal. 22
2 Cosider the followig seteces: " is divisible by 2, or is divisible by 3." " is divisible by 2, ad is divisible by 3." "f the greatest commo divisor of ad 6 is ot, the is divisible by 2, or is divisible 3." " is divisible by 6 if ad oly if is divisible by 2 ad is divisible by 3." By deotig the setece " is divisible by 2 " by A ; also A " is divisible by 2 "; B " is divisible by 3 ", C " is divisible by 6 ", D " The greatest commo divisor of ad 6 is ", the above seteces may be aalyzed, respectively, as E A B, F A B, G ( D) (A B), H C (A B). 23
3 The truth or falsity of the last four composite seteces deped o the truth-values of their costituets A, B, C ad D. (Of course, the truth-value of each of A, B, C ad D deped o the value of the, which we assume to be a fixed, but uspecified, atural umber.) The depedece of the truth-value of E is exactly accordig to the truth-table give above describig the effect of the operatio of (cojuctio) o the two truth-values. That is to say, if A ad B are both true, so is E A B ; i ay other of the three cases cocerig the values of A ad B : (, ), (, ), ad (, ), the value of A B is false. This correspods to the ordiary use of the coective "ad". The truth-value of F A B is computed accordig to the truth-table for (disjuctio) give above. E.g., if = 6, or if = 2, or if = 3, A B is true; i fact, accordig to the first three lies of that table, i the give order. However, if =, the A B is false; this correspods to the last lie of the table. Notice that disjuctio as we are describig it here is o-exclusive "or" ; a disjuctio is true if, i particular, both disjucts are true. The setece i questio is, i more explicit form, "Either is divisible by 2, or is divisible by 3, or both." Let us ote that i mathematics, "or" (disjuctio) is always iteded as o-exclusive "or". (With exclusive "or", a disjuctio would be true just i case precisely oe disjuct is true.) This may be see e.g. o the setece G that is regarded as beig true, o matter what is. f = 6, the the succedet of the coditioal, A B is true uder the o-exclusive iterpretatio, but ot uder the exclusive oe. The coective of egatio as used i mathematical laguage, clearly correspods to the table give for it above. f = 5, the D (with D the setece deoted by D above) is true, ad D is false; if = 2, the D is false, ad D is true. The coective of the coditioal also correspods to a operatio i the two-elemet algebra 2 as follows: =, =, =, =. 24
4 This table says that a coditioal is true uless the atecedet is true, ad the succedet is false. particular, the coditioal is true wheever the atecedet is false, idepedetly of the truth-value of the succedet: "false implies everythig". We may verify that this correspods to the usual mathematical use by cosiderig that the setece that is, C A, "f is divisible by 6, the is divisible by 2 ", should be true o matter what the value of is. f = 6, = 2, =, we obtai the seteces "f 6 is divisible by 6, the 6 is divisible by 2 ", "f 2 is divisible by 6, the 2 is divisible by 2 ", "f is divisible by 6, the is divisible by 2 ". These are of the respective forms,,. As said, ordiary mathematical usage attributes the value true to these forms, i agreemet with the table for the coditioal above. The fact that a coditioal is true oce the atecedet is false is also reflected i the geeral approach to the proof of a coditioal, which is that we start by assumig that the atecedet is true. fact, we may just as well do so, sice if the atecedet is false, the whole coditioal is automatically true, ad we ca rest i our task of provig the coditioal to be true. 25
5 The coditioal ca be expressed i terms of egatio ad disjuctio: x y = (-x) y () is a idetity true for ay values of x ad y i 2 (verify!). Thus, i priciple, the coditioal could be dispesed with; setece G may be paraphrased as "Either the greatest commo divisor of ad 6 is, or is divisible by 2, or is divisible by 3." The bicoditioal has the followig truth-table: = = = =. other words, the bicoditioal is true just i case its terms have equal truth-values. The bicoditioal ca also be expressed i terms of previous coectives: x y = (x y) (y x) (2) (verify!). fact, this correspods to our geeral attitude towards the proof of a bicoditioal, which is that it ivolves the proof of two coditioals. The equalities (), (2) may be cosidered as iitios of the coditioal ad the bicoditioal as operatios i a arbitrary Boolea algebra. case that algebra is (B), the power-set algebra, the, for sets X ad Y B, we have that ad X Y = (-X) Y X Y = ((-X) Y) ((-Y) X). 26
6 Next, we itroduce a geeral costructio o Boolea algebras. Let = (A, ) be ay Boolea algebra, ay set. We cosider all fuctios from ito A as the elemets of a ew Boolea algebra deoted ; read " -to-the-power- ", or more simply, " -to- ". The uderlyig set of is, as we said, A, the set of all * fuctios ξ: A. The orderig i,, is ied compoetwise from : for ξ, ζ A, * ξ ζ ξ(i) ζ(i) for all i. * There are several thigs to check: firstly, that is ideed a order o A ; further, that this order has all the requisite properties to make =(A, * ) a Boolea algebra. fact, what * * * * * happes is that the Boolea operatios,,,, ad - i are all computed compoetwise: for all ξ, ζ A ad i, we have: * (i) =, * (i) =, (ξ ζ)(i) = ξ(i) ζ(i) * (we should have writte ξ ζ, but it is ot ecessary to be that pedatic...). (ξ ζ)(i) = ξ(i) ζ(i) (-ξ)(i) = -(ξ(i)) The proof of all these assertios is easy. For istace, the assertio for is that the fuctio η A for which η(i) = ξ(i) ζ(i) for all i is, i fact, the meet of ξ ad ζ i. Accordig to a display o page 80 i Sectio 3.2, the best way to prove this is showig that for all χ A, * χ η χ ξ ad χ ζ. 27
7 Whe we put i the iitio of η ad that of *, we get for all χ A, χ(i) ξ(i) ζ(i) χ(i) ξ(i) ad χ(i) ζ(i), which, for each i, is a istace of the same relatio o page 80 i Sectio 3.2 for the origial Boolea algebra. Let us apply the power-costructio to the algebra =2. The elemets of the Boolea algebra 2 are the fuctios {, } ; for ξ, ηε{, }, ξ η iff ξ(i) η(i) for all iε. Also ote that sice 2 has just two elemets ad, ad <, ξ(i) η(i) is equivalet to sayig that if ξ(i) =, the η(i) =. The power-algebra 2 () : is i fact a very familiar oe: it is isomorphic to the power-set algebra () 2. Let us specify the isomorphism, i fact, i both directios: f () 2 :: g for X ε (), f(x) is the fuctio {, } for which if u ε X f(x)(u) = if u X ad for ay fuctio ξ ε {, }, g(ξ) is the subset of give as g(ξ) = {uε ξ(u) = }. 28
8 These mappigs f ad g respect the orders, ad they are iverses of each other; these facts are easily checked (exercises). other words, f is a isomorphism f: () 2. Note that, for X (), f(x) is what we call the characteristic fuctio of X. This represetatio of power-set algebras provides a direct proof that ay idetity that holds i the 2-elemet algebra 2 holds i ay power-set algebra, ad hece, i ay Boolea algebra whatsoever. The reaso is that, as it is see by ispectio, a idetity that holds i a algebra holds also i a power of it; also remember that we said i the previous sectio that all idetities i set-algebras hold i all Boolea algebras. See i the light of the last statemet, the iitio of "Boolea algebra" is just a summary of what idetities hold i the algebra of the two truth-values! Note carefully that, if we take this "iitio" of "Boolea algebra" as basic, it is ot obvious -- although ow kow by us -- that the Boolea laws are all cosequeces of the few that we earlier explicitly specified as the Boolea laws. Whe people talk about "Boolea fuctios", they mea fuctios of possibly several variables, all of which rage over the set {, }, ad whose values are also i {, }. (Very ofte (specially i computer sciece), we write for, ad 0 for ; but we will stick to the, -otatio.) f the fuctio f i questio has variables P,..., P [we write P for the variables, sice they are see as "propositios"; i fact, they simply take the values ad ], the f is a fuctio f : {, } {, } ; for ay P,..., P {, }, f(p,..., P ) is agai a elemet of {, }. (2 ) With the fixed, ote that the umber of distict -variable Boolea fuctios is 2. A Boolea fuctio f ca be represeted by a truth-table listig all possible systems of argumet-values, ad the correspodig fuctio-values. For istace, whe =3, the 29
9 truth-table might be like this (we write P, Q, R for P, P, P ): 2 3 P Q R f(p, Q, R) (3) Now, regard the set {, } as the uderlyig set of 2. The the -variable Boolea fuctios form the uderlyig set of the power-algebra 2, where ={, }. other words, for a fixed, the -variable Boolea fuctios themselves form a Boolea algebra. Let us write P to abbreviate P,..., P. The Boolea operatios o the power-algebra ({, 2 } ), the Boolea algebra of -variable Boolea fuctios, is ied compoetwise: (f g)(p) = f(p) g(p), ad similarly for the other operatios. ({, We write, more simply, BF[] for 2 } ), the Boolea algebra of -variable Boolea fuctios. The most atural examples for Boolea fuctios are the Boolea polyomials: these are the fuctios that ca be writte dow by repeated use of the basic Boolea operatios. For istace, whe =3, the followig are Boolea polyomials : ( (((P R) (Q R)) )) R (R Q). 30
10 The fact that the last does ot cotai the variable P does ot make it illegitimate as a three-variable polyomial: this oe simply does ot deped o P. Boolea polyomials should be see as aalogs to ordiary (algebraic) polyomials. The differeces are that, Boolea polyomials are fuctios o the truth-values, istead of umbers; ad the basic Boolea operatios figure i them, istead of the ordiary arithmetical operatios +,, etc. A Boolea polyomial is (or, deotes) a Boolea fuctio: substitutig iite truth-values for the variables, ad usig the basic Boolea operatios o truth-values, we get a iite value for the polyomial. For istace, here are all the values of the first of the two listed polyomials: P Q R ( (((P R) (Q R)) )) R Calculatig the value, for istace i the third lie, looks like this: ( (((P R) (Q R)) )) R this, first, we wrote the value of every variable uder each occurrece of the variable, icludig the value uder the costat i the polyomial; ext, we proceeded to calculate the values of the part-expressios from the iside out; there are as may as there are coectives, occurreces of,, ad -. The umbers idicate the order i which we go through all costituet expressios util we reach the total expressio i stage 8 ; the fial result is that above 8,. There is a slight ambiguity i the meaig of the expressio "Boolea polyomial". We 3
11 sometimes mea the formal expressio itself, rather tha the fuctio deoted by it. However, the official meaig should remai the fuctio itself; whe oe wats to refer to the formal otio, oe should say "formal polyomial". This remark is relevat i the light of the fact that two formally differet Boolea polyomials may be equal to each other. the first of the last two examples, the values i the value-colum coicide with the values of R ; the polyomial coicides (deotes the same fuctio as) the simple polyomial ("moomial") R. Of course, this pheomeo is familiar i the case of ordiary (algebraic) polyomials. E.g, 2 2 the two formal polyomial expressios (x-y)(x+y) ad x -y deote the same polyomial. We ca see this by usig the basic algebraic laws. The situatio with Boolea polyomials is similar. stead of goig through the tables of values (which ted to be very large eve with a moderate umber of variables), we may use the Boolea idetities to establish that two formal Boolea polyomials are the same polyomial. For istace, i the example at had: ( (((P R) (Q R)) )) R = ( (((P Q) R) )) R (distributive law) = ( ((P Q) R) ) R (uit law) = ( (P Q) R) ) R (De Morga) = ( (P Q) R) ) R (double egatio) = R (commutative law, absorptio) fact, what we said about all idetities of Boolea algebras beig true i 2 meas that every time two formal Boolea polyomials are the same fuctio o the truthc-values, this fact ca be deduced by usig the Boolea idetities aloe. Note that the Boolea operatios o the Boolea polyomials as Boolea fuctios are performed by formally applyig the operatio i questio. For istace, if the three variable polyomials metioed above are briefly called f ad g, the f g is (( (((P R) (Q R)) )) R ) (R Q). What this meas is that 32
12 the Boolea polyomials form a subalgebra of BF[]. Cosider ow the variables P, P 2,... P themselves as such Boolea fuctios, i fact, Boolea polyomials. P i is the fuctio that satisfies P (ε,..., ε,..., ε ) = ε ; i i i here, each ε,, ε is a truth-value, or. (This is similar as whe the sigle variable say y is regarded as oe of the ordiary polyomials i variables x, y, z.) We clearly have that the particular elemets P i polyomials. of BF[] geerate the Boolea subalgebra of Boolea Now, claim that the Boolea fuctios P, P 2,... P are idepedet i the Boolea algebra of all -elemet Boolea fuctios. What we have to see is that, for ay distributio of the values ε,, ε i {, }, the meet-expressio ε P ε P... ε P 2 2 is differet from i the Boolea algebra of Boolea fuctios; here, εp meas P if ε =, ad -P if ε =. But if we give the value ε to P, we get that ε P takes i i i i the value : (P)(P= ) = ; (-P)(P= ) = ; thus, (ε P ε P... ε P )(P =ε,..., P =ε ) =... =. 2 2 Sice the fuctio takes the value the -fuctio, which is costat. at at least oe system of argumets, the fuctio is ot Remember that a idepedet family of elemets geerate a Boolea subalgebra of size (2 ) (2 2. t follows that there are exactly 2 ) distict Boolea polyomials. But the 33
13 (2 whole algebra BF[] is of the same size, 2 ). t follows that all Boolea fuctios are (represeted by) Boolea polyomials. fact, all Boolea fuctios ca be writte as jois of complete meet expressios i terms of the variables. The expressios P ad P (or, P ) are also called literals. The expressio of a Boolea i i i fuctio as a joi of distict complete meets of literals is called the disjuctive ormal form (df) of the fuctio. We have that every fuctio has a uique df: the complete meets of literals appearig i the df are determied as those atoms of BF[] that are below the give fuctio. Applyig duality, oe also gets a cojuctive ormal form. The last-stated fact has cocerig the existece of the df also has a direct proof, together with a simple method of producig the df of a fuctio, based o the truth-table of the fuctio. The result is this. Cosider the truth-table of the -variable Boolea fuctio f. Select those lies i the table i which the value of f is case are, 2,..., k. ; say the lies i which this is the Each lie j ( j=,..., k ) has a certai system of the values of the variables. Let us deote the value of P i lie i j by ε ji (ad ot ε ij, because j deotes the k "row-umber", i the "colum-umber"). The df of f is ε ji P i ; or i more j=i= detail, ε P ε 2 P 2... ε P ε 2 P ε 22 P 2... ε 2 P... ε P ε P... ε P. k k2 2 k Here we used the covetio applied before: P is P, P is P. To give a example, cosider the fuctio whose truth-table is (3). There are five lies where 34
14 the value is. The df is PQR PQR PQR PQR PQR. The proof that this is a correct procedure has to show that the df costructed assumes the same values at each system of values for the variables. Now, the df is if ad oly if oe of its disjucts is. But each disjuct correspods to a lie, say j, where the value of f is. The correspodig disjuct is ε P ε P... ε P, (4) j j2 2 j ad this will take the value iff each cojuct ε P takes the value ji i, which is the case if ad oly if P i takes the value ε ji. This meas that the uique system of truth-values where the value of (4) is is precisely the oe i lie j! We have cocluded that the df takes the value exactly i the lies j, for j=,..., k, which are also exactly the lies where f is. This proves that the df ad f are idetical fuctios. The df of a Boolea fuctio may be extremely large already i case of a moderate umber of variables. The problem of Boolea realizatio is to fid a possibly small formal Boolea polyomial represetig a give Boolea fuctio. 35
6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationLecture Notes for CS 313H, Fall 2011
Lecture Notes for CS 313H, Fall 011 August 5. We start by examiig triagular umbers: T () = 1 + + + ( = 0, 1,,...). Triagular umbers ca be also defied recursively: T (0) = 0, T ( + 1) = T () + + 1, or usig
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationResolution Proofs of Generalized Pigeonhole Principles
Resolutio Proofs of Geeralized Pigeohole Priciples Samuel R. Buss Departmet of Mathematics Uiversity of Califoria, Berkeley Győrgy Turá Departmet of Mathematics, Statistics, ad Computer Sciece Uiversity
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationMathematical Induction
Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
More informationRelations Among Algebras
Itroductio to leee Algebra Lecture 6 CS786 Sprig 2004 February 9, 2004 Relatios Amog Algebras The otio of free algebra described i the previous lecture is a example of a more geeral pheomeo called adjuctio.
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More information10. Comparative Tests among Spatial Regression Models. Here we revisit the example in Section 8.1 of estimating the mean of a normal random
Part III. Areal Data Aalysis 0. Comparative Tests amog Spatial Regressio Models While the otio of relative likelihood values for differet models is somewhat difficult to iterpret directly (as metioed above),
More informationChapter IV Integration Theory
Chapter IV Itegratio Theory Lectures 32-33 1. Costructio of the itegral I this sectio we costruct the abstract itegral. As a matter of termiology, we defie a measure space as beig a triple (, A, µ), where
More informationThe multiplicative structure of finite field and a construction of LRC
IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationBasic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.
Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationStructural Functionality as a Fundamental Property of Boolean Algebra and Base for Its Real-Valued Realizations
Structural Fuctioality as a Fudametal Property of Boolea Algebra ad Base for Its Real-Valued Realizatios Draga G. Radojević Uiversity of Belgrade, Istitute Mihajlo Pupi, Belgrade draga.radojevic@pupi.rs
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationCS284A: Representations and Algorithms in Molecular Biology
CS284A: Represetatios ad Algorithms i Molecular Biology Scribe Notes o Lectures 3 & 4: Motif Discovery via Eumeratio & Motif Represetatio Usig Positio Weight Matrix Joshua Gervi Based o presetatios by
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationMath F215: Induction April 7, 2013
Math F25: Iductio April 7, 203 Iductio is used to prove that a collectio of statemets P(k) depedig o k N are all true. A statemet is simply a mathematical phrase that must be either true or false. Here
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More information2 Geometric interpretation of complex numbers
2 Geometric iterpretatio of complex umbers 2.1 Defiitio I will start fially with a precise defiitio, assumig that such mathematical object as vector space R 2 is well familiar to the studets. Recall that
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationExercises 1 Sets and functions
Exercises 1 Sets ad fuctios HU Wei September 6, 018 1 Basics Set theory ca be made much more rigorous ad built upo a set of Axioms. But we will cover oly some heuristic ideas. For those iterested studets,
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationSome examples of vector spaces
Roberto s Notes o Liear Algebra Chapter 11: Vector spaces Sectio 2 Some examples of vector spaces What you eed to kow already: The te axioms eeded to idetify a vector space. What you ca lear here: Some
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationLecture 2. The Lovász Local Lemma
Staford Uiversity Sprig 208 Math 233A: No-costructive methods i combiatorics Istructor: Ja Vodrák Lecture date: Jauary 0, 208 Origial scribe: Apoorva Khare Lecture 2. The Lovász Local Lemma 2. Itroductio
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More information1. ARITHMETIC OPERATIONS IN OBSERVER'S MATHEMATICS
1. ARITHMETIC OPERATIONS IN OBSERVER'S MATHEMATICS We cosider a ite well-ordered system of observers, where each observer sees the real umbers as the set of all iite decimal fractios. The observers are
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationSquare-Congruence Modulo n
Square-Cogruece Modulo Abstract This paper is a ivestigatio of a equivalece relatio o the itegers that was itroduced as a exercise i our Discrete Math class. Part I - Itro Defiitio Two itegers are Square-Cogruet
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationa 2 +b 2 +c 2 ab+bc+ca.
All Problems o the Prize Exams Sprig 205 The source for each problem is listed below whe available; but eve whe the source is give, the formulatio of the problem may have bee chaged. Solutios for the problems
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,
More informationModel Theory 2016, Exercises, Second batch, covering Weeks 5-7, with Solutions
Model Theory 2016, Exercises, Secod batch, coverig Weeks 5-7, with Solutios 3 Exercises from the Notes Exercise 7.6. Show that if T is a theory i a coutable laguage L, haso fiite model, ad is ℵ 0 -categorical,
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationMath 299 Supplement: Real Analysis Nov 2013
Math 299 Supplemet: Real Aalysis Nov 203 Algebra Axioms. I Real Aalysis, we work withi the axiomatic system of real umbers: the set R alog with the additio ad multiplicatio operatios +,, ad the iequality
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationMath 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationsubcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016
subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationThe Borel hierarchy classifies subsets of the reals by their topological complexity. Another approach is to classify them by size.
Lecture 7: Measure ad Category The Borel hierarchy classifies subsets of the reals by their topological complexity. Aother approach is to classify them by size. Filters ad Ideals The most commo measure
More informationR is a scalar defined as follows:
Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More information2.4 Sequences, Sequences of Sets
72 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.4 Sequeces, Sequeces of Sets 2.4.1 Sequeces Defiitio 2.4.1 (sequece Let S R. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For each
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationThe Growth of Functions. Theoretical Supplement
The Growth of Fuctios Theoretical Supplemet The Triagle Iequality The triagle iequality is a algebraic tool that is ofte useful i maipulatig absolute values of fuctios. The triagle iequality says that
More informationLONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES
J Lodo Math Soc (2 50, (1994, 465 476 LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES Jerzy Wojciechowski Abstract I [5] Abbott ad Katchalski ask if there exists a costat c >
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationProblem Set 2 Solutions
CS271 Radomess & Computatio, Sprig 2018 Problem Set 2 Solutios Poit totals are i the margi; the maximum total umber of poits was 52. 1. Probabilistic method for domiatig sets 6pts Pick a radom subset S
More information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More informationOnce we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1
. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More information(for homogeneous primes P ) defining global complex algebraic geometry. Definition: (a) A subset V CP n is algebraic if there is a homogeneous
Math 6130 Notes. Fall 2002. 4. Projective Varieties ad their Sheaves of Regular Fuctios. These are the geometric objects associated to the graded domais: C[x 0,x 1,..., x ]/P (for homogeeous primes P )
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More information11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.
11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationDeterminants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1)
5. Determiats 5.. Itroductio 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios 5.3. A Set of Axioms for a Determiat Fuctio 5.4. The Determiat of a Diagoal Matrix 5.5. The Determiat of a Upper
More informationThe Structure of Z p when p is Prime
LECTURE 13 The Structure of Z p whe p is Prime Theorem 131 If p > 1 is a iteger, the the followig properties are equivalet (1) p is prime (2) For ay [0] p i Z p, the equatio X = [1] p has a solutio i Z
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More informationLecture XVI - Lifting of paths and homotopies
Lecture XVI - Liftig of paths ad homotopies I the last lecture we discussed the liftig problem ad proved that the lift if it exists is uiquely determied by its value at oe poit. I this lecture we shall
More informationProof of Goldbach s Conjecture. Reza Javaherdashti
Proof of Goldbach s Cojecture Reza Javaherdashti farzijavaherdashti@gmail.com Abstract After certai subsets of Natural umbers called Rage ad Row are defied, we assume (1) there is a fuctio that ca produce
More informationOn a Smarandache problem concerning the prime gaps
O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationPb ( a ) = measure of the plausibility of proposition b conditional on the information stated in proposition a. & then using P2
Axioms for Probability Logic Pb ( a ) = measure of the plausibility of propositio b coditioal o the iformatio stated i propositio a For propositios a, b ad c: P: Pb ( a) 0 P2: Pb ( a& b ) = P3: Pb ( a)
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationBasics of Probability Theory (for Theory of Computation courses)
Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.
More information