Combinatorics 71 5.3. of connected components in graph with n vertices = n5. | x | = { x if x 0 x if x < 0. The number of such arrangements is given by $C(n, r)$, defined as: Remark: we note that for $0\leqslant r\leqslant n$, we have $P(n,r)\geqslant C(n,r)$. The no. stream By using this website, you agree with our Cookies Policy. Let q = a b and r = c d be two rational numbers written in lowest terms. I go out of my way to simplify subjects. By noting $f$ and $F$ the PDF and CDF respectively, we have the following relations: Continuous case Here, $X$ takes continuous values, such as the temperature in the room. This ordered or stable list of counting words must be at least as long as the number of items to be counted. BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. /Filter /FlateDecode It is determined as follows: Characteristic function A characteristic function $\psi(\omega)$ is derived from a probability density function $f(x)$ and is defined as: Euler's formula For $\theta \in \mathbb{R}$, the Euler formula is the name given to the identity: Revisiting the $k^{th}$ moment The $k^{th}$ moment can also be computed with the characteristic function as follows: Transformation of random variables Let the variables $X$ and $Y$ be linked by some function. Probability density function (PDF) The probability density function $f$ is the probability that $X$ takes on values between two adjacent realizations of the random variable. \newcommand{\vl}[1]{\vtx{left}{#1}} You can use all your notes, calcu-lator, and any books you Different three digit numbers will be formed when we arrange the digits. A combination is selection of some given elements in which order does not matter. Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. of bijection function =n!6. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. /Creator () Simple is harder to achieve. No. I dont know whether I agree with the name, but its a nice cheat sheet. I hate discrete math because its hard for me to understand. For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. Types of propositions based on Truth values1.Tautology A proposition which is always true, is called a tautology.2.Contradiction A proposition which is always false, is called a contradiction.3.Contingency A proposition that is neither a tautology nor a contradiction is called a contingency. /ImageMask true The function is surjective (onto) if every element of the codomain is mapped to by at least one element. \renewcommand{\bar}{\overline} $A \cap B = \emptyset$), then mathematically $|A \cup B| = |A| + |B|$, The Rule of Product If a sequence of tasks $T_1, T_2, \dots, T_m$ can be done in $w_1, w_2, \dots w_m$ ways respectively and every task arrives after the occurrence of the previous task, then there are $w_1 \times w_2 \times \dots \times w_m$ ways to perform the tasks. << How many ways can you choose 3 distinct groups of 3 students from total 9 students? Webdiscrete math counting cheat sheet.pdf - | Course Hero University of California, Los Angeles MATH MATH 61 discrete math counting cheat sheet.pdf - discrete math No. \newcommand{\st}{:} Did you make this project? gQVmDYm*% QKP^n,D%7DBZW=pvh#(sG In a group of 50 students 24 like cold drinks and 36 like hot drinks and each student likes at least one of the two drinks. Minimum number of connected components =, 6. \newcommand{\amp}{&} xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d /Title ( D i s c r e t e M a t h C h e a t S h e e t b y D o i s - C h e a t o g r a p h y . So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. /Resources 23 0 R Partition Let $\{A_i, i\in[\![1,n]\! /Length 530 endobj \newcommand{\lt}{<} \newcommand{\card}[1]{\left| #1 \right|} endobj ~C'ZOdA3,3FHaD%B,e@,*/x}9Scv\`{]SL*|)B(u9V|My\4 Xm$qg3~Fq&M?D'Clk +&$.U;n8FHCfQd!gzMv94NU'M`cU6{@zxG,,?F,}I+52XbQN0.''f>:Vn(g."]^{\p5,`"zI%nO. /N 100 Boolean Lattice: It should be both complemented and distributive. For instance, in how many ways can a panel of judges comprising of 6 men and 4 women be chosen from among 50 men and 38 women? In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. Pascal's identity, first derived by Blaise Pascal in 17 century, states that \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. <> WebThe first principle of counting involves the student using a list of words to count in a repeatable order. of symmetric relations = 2n(n+1)/29. (1!)(1!)(2!)] Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. The order of elements does not matter in a combination.which gives us-, Binomial Coefficients: The -combinations from a set of elements if denoted by . Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. Rsolution chap02 - Corrig du chapitre 2 de benson Physique 2; CCNA 1 v7 Modules 16 17 Building and Securing a Small Network Exam Answers; Processing and value addition in ornamental flower crops (2019-AJ-66) Chapitre 3 r ponses (STE) Homework 9.3 1 0 obj << stream After filling the first and second place, (n-2) number of elements is left. WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. Hence, a+c b+d(modm)andac bd(modm). Web2362 Education Cheat Sheets. /Length 1235 To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B.To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B. Size of the set S is known as Cardinality number, denoted as |S|. of edges to have connected graph with n vertices = n-17. Here's how they described it: Equations commonly used in Discrete Math. ]8$_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 /SMask /None>> It is determined as follows: Standard deviation The standard deviation of a random variable, often noted $\sigma$, is a measure of the spread of its distribution function which is compatible with the units of the actual random variable. >> endobj %PDF-1.4 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 The number of ways to choose 3 men from 6 men is $^6C_{3}$ and the number of ways to choose 2 women from 5 women is $^5C_{2}$, Hence, the total number of ways is $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. /MediaBox [0 0 612 792] We can also write N+= {x N : x > 0}. /ProcSet [ /PDF /Text ] Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. To guarantee that a graph with n vertices is connected, minimum no. :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X Expected value The expected value of a random variable, also known as the mean value or the first moment, is often noted $E[X]$ or $\mu$ and is the value that we would obtain by averaging the results of the experiment infinitely many times. endobj Below is a quick refresher on some math tools and problem-solving techniques from 240 (or other prereqs) that well assume knowledge of for the PSets. = 180.$. \newcommand{\va}[1]{\vtx{above}{#1}} }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. A graph is euler graph if it there exists atmost 2 vertices of odd degree9. Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. stream on April 20, 2023, 5:30 PM EDT. \newcommand{\R}{\mathbb R} acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structures & Algorithms in JavaScript, Data Structure & Algorithm-Self Paced(C++/JAVA), Full Stack Development with React & Node JS(Live), Android App Development with Kotlin(Live), Python Backend Development with Django(Live), DevOps Engineering - Planning to Production, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Mathematics | Introduction to Propositional Logic | Set 1, Discrete Mathematics Applications of Propositional Logic, Difference between Propositional Logic and Predicate Logic, Mathematics | Predicates and Quantifiers | Set 1, Mathematics | Some theorems on Nested Quantifiers, Mathematics | Set Operations (Set theory), Mathematics | Sequence, Series and Summations, Mathematics | Representations of Matrices and Graphs in Relations, Mathematics | Introduction and types of Relations, Mathematics | Closure of Relations and Equivalence Relations, Permutation and Combination Aptitude Questions and Answers, Discrete Maths | Generating Functions-Introduction and Prerequisites, Inclusion-Exclusion and its various Applications, Project Evaluation and Review Technique (PERT), Mathematics | Partial Orders and Lattices, Mathematics | Probability Distributions Set 1 (Uniform Distribution), Mathematics | Probability Distributions Set 2 (Exponential Distribution), Mathematics | Probability Distributions Set 3 (Normal Distribution), Mathematics | Probability Distributions Set 5 (Poisson Distribution), Mathematics | Graph Theory Basics Set 1, Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph, Mathematics | Independent Sets, Covering and Matching, How to find Shortest Paths from Source to all Vertices using Dijkstras Algorithm, Introduction to Tree Data Structure and Algorithm Tutorials, Prims Algorithm for Minimum Spanning Tree (MST), Kruskals Minimum Spanning Tree (MST) Algorithm, Tree Traversals (Inorder, Preorder and Postorder), Travelling Salesman Problem using Dynamic Programming, Check whether a given graph is Bipartite or not, Eulerian path and circuit for undirected graph, Fleurys Algorithm for printing Eulerian Path or Circuit, Chinese Postman or Route Inspection | Set 1 (introduction), Graph Coloring | Set 1 (Introduction and Applications), Check if a graph is Strongly, Unilaterally or Weakly connected, Handshaking Lemma and Interesting Tree Properties, Mathematics | Rings, Integral domains and Fields, Topic wise multiple choice questions in computer science, A graph is planar if and only if it does not contain a subdivision of K. Let G be a connected planar graph, and let n, m and f denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of G. Then n m + f = 2. Cumulative distribution function (CDF) The cumulative distribution function $F$, which is monotonically non-decreasing and is such that $\underset{x\rightarrow-\infty}{\textrm{lim}}F(x)=0$ and $\underset{x\rightarrow+\infty}{\textrm{lim}}F(x)=1$, is defined as: Remark: we have $P(a < X\leqslant B)=F(b)-F(a)$. of edges in a complete graph = n(n-1)/22. From a set S ={x, y, z} by taking two at a time, all permutations are , We have to form a permutation of three digit numbers from a set of numbers $S = \lbrace 1, 2, 3 \rbrace$. Let G be a connected planar simple graph with n vertices, where n ? See Last Minute Notes on all subjects here. Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. \newcommand{\Iff}{\Leftrightarrow} No. 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. /SA true 4 0 obj 24 0 obj << endobj 8"NE!OI6%pu=s{ZW"c"(E89/48q << Define the set Ento be the set of binary strings with n bits that have an even number of 1's. ]$, The number of circular permutations of n different elements taken x elements at time = $^np_{x}/x$, The number of circular permutations of n different things = $^np_{n}/n$. Cartesian ProductsLet A and B be two sets. Graphs 82 7.2. Corollary Let m be a positive integer and let a and b be integers. Download the PDF version here. x3T0 BCKs=S\.t;!THcYYX endstream For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. In other words a Permutation is an ordered Combination of elements. Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream Besides, your proof of 0!=1 needs some more attention. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. A permutation is an arrangement of some elements in which order matters.

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