Hence, there are (n-2) ways to fill up the third place. ]$, 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 = $^np_{n}/n$. { (k-1)!(n-k)! } 5 0 obj << << Define P(n) to be the assertion that: j=1nj2=n(n+1)(2n+1)6 (a) Verify that P(3) is true. 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. /ca 1.0 *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! /Creator () x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] We have: Independence Two events $A$ and $B$ are independent if and only if we have: Random variable A random variable, often noted $X$, is a function that maps every element in a sample space to a real line. 28 0 obj << Here, the ordering does not matter. of asymmetric relations = 3n(n-1)/211. /Length 1235 I strongly believe that simple is better than complex. \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. element of the domain. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). x3T0 BCKs=S\.t;!THcYYX endstream 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 Get up and running with ChatGPT with this comprehensive cheat sheet. ]\}$ be such that for all $i$, $A_i\neq\varnothing$. c o m) /MediaBox [0 0 612 792] Hi matt392, nice work! of edges required = {(n-1)*(n-2)/2 } + 18. Prove or disprove the following two statements. If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. /\: [(2!) 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. endobj To guarantee that a graph with n vertices is connected, minimum no. a b. Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$. endobj of one to one function = (n, P, m)3. on Introduction. Assume that s is not 0. There are $50/6 = 8$ numbers which are multiples of both 2 and 3. of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . WebDefinitions. of Anti Symmetric Relations = 2n*3n(n-1)/210. Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. }$$. Graphs 82 7.2. on April 20, 2023, 5:30 PM EDT. WebBefore tackling questions like these, let's look at the basics of counting. Tree, 10. stream (\frac{ k } { k!(n-k)! } Once we can count, we can determine the likelihood of a particular even and we can estimate how long a computer algorithm takes to complete a task. 6 0 obj \newcommand{\vb}[1]{\vtx{below}{#1}} ("#} &. #p Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K Let G be a connected planar simple graph with n vertices, where n ? 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. 3 0 obj Course Hero is not sponsored or endorsed by any college or university. 14 0 obj *3-d[\HxSi9KpOOHNn uiKa, Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). Graph Theory; Notes on Counting; Notes on Distributions and Stirling numbers of the second kind; Notes on Cardinality of Sets; Notes on the Pigeonhole Principle; Notes on Combinatorial Arguments; Notes on Recurrence Relations; Notes on Inclusion-Exclusion; Notes on Generating Functions \newcommand{\inv}{^{-1}} A set A is said to be subset of another set B if and only if every element of set A is also a part of other set B.Denoted by .A B denotes A is a subset of B. A country has two political parties, the Demonstrators and the Repudiators. of edges to have connected graph with n vertices = n-17. 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. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. There are $50/3 = 16$ numbers which are multiples of 3. /SA true Then(a+b)modm= ((amodm) + \newcommand{\pow}{\mathcal P} Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! }$, $= (n-1)! WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. %PDF-1.3 xm=j0 gRR*9BGRGF. I have a class in it right now actually! 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists Hence, the total number of permutation is $6 \times 6 = 36$. Paths and Circuits 91 3 Pascal's Identity. Math/CS cheat sheet. /MediaBox [0 0 612 792] <> In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. How many like both coffee and tea? \newcommand{\st}{:} $A \cap B = \emptyset$), 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. 9 years ago 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID =HA`miNDKH;8&.2_LcVfgsIVAxx$A,t([k9QR$jmOX#Q=s'0z>SUxH-5OPuVq+"a;F} WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. xY8_1ow>;|D@`a%e9l96=u=uQ (c) Express P(k + 1). Representations of Graphs 88 7.3. Equivalesistheonlyequivalencerelationthatisassociative ((p q) r) (p (q of edges =m*n3. [/Pattern /DeviceRGB] Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. You can use all your notes, calcu-lator, and any books you That Let s = q + r and s = e f be written in lowest terms. xKs6. 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. How many ways are there to go from X to Z? <> The cardinality of A B is N*M, where N is the Cardinality of A and M is the cardinality of B. UnionUnion of the sets A and B, denoted by A B, is the set of distinct element belongs to set A or set B, or both. I dont know whether I agree with the name, but its a nice cheat sheet. 6 0 obj That's a good collection you've got there, but your typesetting is aweful, I could help you with that. No. /ProcSet [ /PDF ] xWn7Wgv 23 0 obj << CS160 - Fall Semester 2015. For solving these problems, mathematical theory of counting are used. % /Contents 3 0 R 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 . 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} xS@}WD"f<7.\$.iH(Rc'vbo*g1@9@I4_ F2 }3^C2>2B@>8JfWkn%;?t!yb C;.AIyir!zZn}Na;$t"2b {HEx}]Zg;'B!e>3B=DWw,qS9\ THi_WI04$-1cb By noting $f_X$ and $f_Y$ the distribution function of $X$ and $Y$ respectively, we have: Leibniz integral rule Let $g$ be a function of $x$ and potentially $c$, and $a, b$ boundaries that may depend on $c$. For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? %PDF-1.4 Generalized Permutations and Combinations 73 5.4. in the word 'READER'. Event Any subset $E$ of the sample space is known as an event. + \frac{ n-k } { k!(n-k)! } Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. We can also write N+= {x N : x > 0}. `y98R uA>?2 AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! No. The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. /Type /Page Hence, there are 10 students who like both tea and coffee. Discrete Math Cheat Sheet by Dois via cheatography.com/11428/cs/1340/ Complex Numbers j = -1 j = -j j = 1 z = a + bj z = r(sin + jsin) z = re tan b/a = A cos a/r In this case the sign means that a divides b, or that b a is an integer. A relation is an equivalence if, 1. << So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } \newcommand{\imp}{\rightarrow} }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. \newcommand{\vr}[1]{\vtx{right}{#1}} (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! One of the first things you learn in mathematics is how to count. )$. In this case it is written with just the | symbol. / [(a_1!(a_2!) Then, The binomial expansion using Combinatorial symbols. How many anagrams are there of anagram? /Subtype /Image Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. The number of such arrangements is given by $P(n, r)$, defined as: Combination A combination is an arrangement of $r$ objects from a pool of $n$ objects, where the order does not matter. U denotes the universal set. 1.Implication : 2.Converse : The converse of the proposition is 3.Contrapositive : The contrapositive of the proposition is 4.Inverse : The inverse of the proposition is. 17 0 obj So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. Thank you - hope it helps. n Less theory, more problem solving, focuses on exam problems, use as study sheet! WebProof : Assume that n is an odd integer. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe 9Wr\Gux[Eul\? gQVmDYm*% QKP^n,D%7DBZW=pvh#(sG Number of permutations of n distinct elements taking n elements at a time = $n_{P_n} = n!$, The number of permutations of n dissimilar elements taking r elements at a time, when x particular things always occupy definite places = $n-x_{p_{r-x}}$, The number of permutations of n dissimilar elements when r specified things always come together is $r! Minimum number of connected components =, 6. Partition Let $\{A_i, i\in[\![1,n]\! Affordable solution to train a team and make them project ready. No. Probability 78 6.1. \renewcommand{\iff}{\leftrightarrow} Above Venn Diagram shows that A is a subset of B. Prove the following using a proof by contrapositive: Let x be a rational number. /CreationDate (D:20151115165753Z) A poset is called Lattice if it is both meet and join semi-lattice16. Note that zero is an even number, so a string. /Type /ObjStm We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. WebDiscrete and Combinatorial Mathematics. on Introduction.