discrete math counting cheat sheet

Discrete Mathematics - Counting Theory 1 The Rules of Sum and Product. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. 2 Permutations. A permutation is an arrangement of some elements in which order matters. 3 Combinations. 4 Pascal's Identity. 5 Pigeonhole Principle. << Rsolution chap02 - Corrig du chapitre 2 de benson Physique 2; CCNA 1 v7 Modules 16 17 Building and Securing a Small Network Exam Answers; Processing and value addition in ornamental flower crops (2019-AJ-66) Chapitre 3 r ponses (STE) Homework 9.3 Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional on Introduction. ]8$_v'6\2V1A) cz^U@2"jAS?@nF'8C!g1ZF%54fI4HIs e"@hBN._4~[E%V?#heH1P|'?0D#jX4Ike+{7fmc"Y$c1Fj%OIRr2^0KS)6,u`k*2D8X~@ @49d)S!Y+ad~T3=@YA )w[Il35yNrk!3PdsoZ@iqFd39|x;MUqK.-DbV]kx7VqD[h6Y[r]sd}?%endstream What helped me was to take small bits of information and write them out 25 times or so. Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, recursive formulas, summation, logic, sets, power sets, functions, combinatorics, arrays and matrices. Did you make this project? Share it with us! I Made It! on April 20, 2023, 5:30 PM EDT. %PDF-1.2 Here it means the absolute value of x, ie. ];_. :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx Equivalesistheonlyequivalencerelationthatisassociative ((p q) r) (p (q xKs6. Paths and Circuits 91 3 stream Hence, there are (n-1) ways to fill up the second place. 9 years ago o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X WebDiscrete Mathematics Cheat Sheet Set Theory Definitions Set Definition:A set is a collection of objects called elements Visual Representation: 1 2 3 List Notation: {1,2,3} Discrete Mathematics Cheat Sheet - DocDroid Now, it is known as the pigeonhole principle. For choosing 3 students for 1st group, the number of ways $^9C_{3}$, The number of ways for choosing 3 students for 2nd group after choosing 1st group $^6C_{3}$, The number of ways for choosing 3 students for 3rd group after choosing 1st and 2nd group $^3C_{3}$, Hence, the total number of ways $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. \renewcommand{\iff}{\leftrightarrow} Binomial Coecients 75 5.5. Size of a SetSize of a set can be finite or infinite. For instance, in how many ways can a panel of judges comprising of 6 men and 4 women be chosen from among 50 men and 38 women? %PDF-1.4 = 6$. | x |. Discrete Structures Lecture Notes - Stanford University FWfSE xpwy8+3o WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. Probability For Dummies Cheat Sheet - dummies Counting problems may be hard, and easy solutions are not obvious Approach: simplify the solution by decomposing the problem Two basic decomposition rules: Product rule A count decomposes into a sequence of dependent counts (each element in the first count is associated with all elements of the second count) Sum rule From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. \dots (a_r!)]$. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. /N 100 Learn everything from how to sign up for free to enterprise WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. \(\renewcommand{\d}{\displaystyle} Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? <> %PDF-1.3 CS160 - Fall Semester 2015. 5 0 obj << The order of elements does not matter in a combination.which gives us-, Binomial Coefficients: The -combinations from a set of elements if denoted by . Harold's Cheat Sheets "If you can't explain it simply, you don't understand it well enough." Course Hero is not sponsored or endorsed by any college or university. Note that zero is an even number, so a string. ~C'ZOdA3,3FHaD%B,e@,*/x}9Scv\`{]SL*|)B(u9V|My\4 Xm$qg3~Fq&M?D'Clk +&$.U;n8FHCfQd!gzMv94NU'M`cU6{@zxG,,?F,}I+52XbQN0.''f>:Vn(g."]^{\p5,`"zI%nO. /Type /Page \newcommand{\lt}{<} /SM 0.02 WebDiscrete and Combinatorial Mathematics. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE Prove or disprove the following two statements. So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. No. `y98R uA>?2 AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! How many ways are there to go from X to Z? WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . Event Any subset $E$ of the sample space is known as an event. It is computed as follows: Generalization of the expected value The expected value of a function of a random variable $g(X)$ is computed as follows: $k^{th}$ moment The $k^{th}$ moment, noted $E[X^k]$, is the value of $X^k$ that we expect to observe on average on infinitely many trials. Proof : Assume that m and n are both squares. You can use all your notes, calcu-lator, and any books you 9 years ago /Contents 25 0 R = 720$. Thus, n2 is odd. Probability 78 6.1. 23 0 obj << 1 0 obj It includes the enumeration or counting of objects having certain properties. Discrete mathematics cheat sheet <> of the domain. \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. Size of the set S is known as Cardinality number, denoted as |S|. Axioms of probability For each event $E$, we denote $P(E)$ as the probability of event $E$ occurring. 5 0 obj in the word 'READER'. Toomey.org Tutoring Resources \newcommand{\B}{\mathbf B} Before tackling questions like these, let's look at the basics of counting. By noting $f$ and $F$ the PDF and CDF respectively, we have the following relations: In the following sections, we are going to keep the same notations as before and the formulas will be explicitly detailed for the discrete (D) and continuous (C) cases. 445 Cheatsheet - Princeton University xmT;s1Wli+,[-:^Q1GL$E=>]KC}{~=ogwh=9-} }pNY@z }>c? A combination is selection of some given elements in which order does not matter. One of the first things you learn in mathematics is how to count. 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 Web2362 Education Cheat Sheets. of edges in a complete graph = n(n-1)/22. /Type /ObjStm endobj >> endobj /Length 1781 #p Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K % ("#} &. A graph is euler graph if it there exists atmost 2 vertices of odd degree9. WebDefinitions. Problem 2 In how many ways can the letters of the word 'READER' be arranged? 3 0 obj 14 0 obj \newcommand{\inv}{^{-1}} Probability Cheatsheet v2.0 Thinking Conditionally Law of Definitions // Set A contains elements 1,2 and 3 A = {1,2,3} @>%c0xC8a%k,s;b !AID/~ The cardinality of the set is 6 and we have to choose 3 elements from the set. >> Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. Cartesian ProductsLet A and B be two sets. /Parent 22 0 R DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. this looks promising :), Reply We make use of First and third party cookies to improve our user experience. Combinatorics 71 5.3. Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$.

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