Discrete Math Cram Sheet - Ateneo de Manila University Affordable solution to train a team and make them project ready. To guarantee that a graph with n vertices is connected, minimum no. /Height 25 /First 812 endobj \renewcommand{\iff}{\leftrightarrow} Now we want to count large collections of things quickly and precisely. \newcommand{\gt}{>} Different three digit numbers will be formed when we arrange the digits. FWfSE xpwy8+3o Discrete Mathematics How many like both coffee and tea? We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. `y98R uA>?2 AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! 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? >> If there are n elements of which $a_1$ are alike of some kind, $a_2$ are alike of another kind; $a_3$ are alike of third kind and so on and $a_r$ are of $r^{th}$ kind, where $(a_1 + a_2 + a_r) = n$. on April 20, 2023, 5:30 PM EDT. 592 Graphs 82 7.2. The permutation will be $= 6! <> 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 Cheat Sheet So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. It includes the enumeration or counting of objects having certain properties. Here, the ordering does not matter. Counting Principles - Counting and Cardinality /Decode [1 0] Proof Let there be n different elements. Problem 1 From a bunch of 6 different cards, how many ways we can permute it? How many integers from 1 to 50 are multiples of 2 or 3 but not both? Binomial Coecients 75 5.5. We have: Covariance We define the covariance of two random variables $X$ and $Y$, that we note $\sigma_{XY}^2$ or more commonly $\textrm{Cov}(X,Y)$, as follows: Correlation By noting $\sigma_X, \sigma_Y$ the standard deviations of $X$ and $Y$, we define the correlation between the random variables $X$ and $Y$, noted $\rho_{XY}$, as follows: Remark 1: we note that for any random variables $X, Y$, we have $\rho_{XY}\in[-1,1]$. 8"NE!OI6%pu=s{ZW"c"(E89/48q \renewcommand{\bar}{\overline} WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <> Generalized Permutations and Combinations 73 5.4. No. a b. Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. 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! Pascal's identity, first derived by Blaise Pascal in 17 century, states that That (c) Express P(k + 1). There must be at least two people in a class of 30 whose names start with the same alphabet. The cardinality of the set is 6 and we have to choose 3 elements from the set. 1 0 obj << Web2362 Education Cheat Sheets. By using our site, you U denotes the universal set. /Type /Page Hence, there are (n-1) ways to fill up the second place. The Rule of Sum 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 (the condition is that no tasks can be performed simultaneously), then the number of ways to do one of these tasks is $w_1 + w_2 + \dots +w_m$. After filling the first place (n-1) number of elements is left. 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. WebProof : Assume that n is an odd integer. *3-d[\HxSi9KpOOHNn uiKa, 6 0 obj /ImageMask true I go out of my way to simplify subjects. Here it means the absolute value of x, ie. 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 Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. @>%c0xC8a%k,s;b !AID/~ This ordered or stable list of counting words must be at least as long as the number of items to be counted. of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. Heres something called a theoretical computer science cheat sheet. << Counting Discrete Mathematics Cheat Sheet - DocDroid It wasn't meant to be a presentation per se, but more of a study sheet, so I did not work too hard on the typesetting. | x | = { x if x 0 x if x < 0. A Set is an unordered collection of objects, known as elements or members of the set.An element a belong to a set A can be written as a ∈ A, a A denotes that a is not an element of the set A. 5 0 obj If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. of irreflexive relations = 2n(n-1), 15. Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. /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 . of reflexive relations =2n(n-1)8. Cartesian product of A and B is denoted by A B, is the set of all ordered pairs (a, b), where a belong to A and b belong to B. endobj So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. /Parent 22 0 R Minimum number of connected components =, 6. Graph Theory 82 7.1. Cheat Sheet of Mathemtical Notation and Terminology 1 0 obj \newcommand{\U}{\mathcal U} endobj Download the PDF version here. 3 0 obj For $k, \sigma>0$, we have the following inequality: Discrete distributions Here are the main discrete distributions to have in mind: Continuous distributions Here are the main continuous distributions to have in mind: Joint probability density function The joint probability density function of two random variables $X$ and $Y$, that we note $f_{XY}$, is defined as follows: Marginal density We define the marginal density for the variable $X$ as follows: Cumulative distribution We define cumulative distrubution $F_{XY}$ as follows: Conditional density The conditional density of $X$ with respect to $Y$, often noted $f_{X|Y}$, is defined as follows: Independence Two random variables $X$ and $Y$ are said to be independent if we have: Moments of joint distributions We define the moments of joint distributions of random variables $X$ and $Y$ as follows: Distribution of a sum of independent random variables Let $Y=X_1++X_n$ with $X_1, , X_n$ independent. %PDF-1.4 Hence, the number of subsets will be $^6C_{3} = 20$. Then m 3n 6. /SM 0.02 Thank you - hope it helps. Equivalesistheonlyequivalencerelationthatisassociative ((p q) r) (p (q \(\renewcommand{\d}{\displaystyle} % 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. Note that in this case it is written \mid in LaTeX, and not with the symbol |. \definecolor{fillinmathshade}{gray}{0.9} The function is injective (one-to-one) if every element of the codomain is mapped to by at most one. 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} endobj /SA true Ten men are in a room and they are taking part in handshakes. stream /MediaBox [0 0 612 792] Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. '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} on April 20, 2023, 5:30 PM EDT. WebBefore tackling questions like these, let's look at the basics of counting. \newcommand{\C}{\mathbb C} 5 0 obj << \newcommand{\R}{\mathbb R} Counting - Discrete Mathematics Basic rules to master beginner French! /N 100 x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] 28 0 obj << of spanning tree possible = nn-2. Probability 78 Chapter 7. Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. 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. From a night class at Fordham University, NYC, Fall, 2008. WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. 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 $c$. xWn7Wgv \newcommand{\Iff}{\Leftrightarrow} \newcommand{\isom}{\cong} DISCRETE MATHEMATICS FOR COMPUTER SCIENCE % I dont know whether I agree with the name, but its a nice cheat sheet. 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. The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! \newcommand{\vl}[1]{\vtx{left}{#1}} \dots (a_r!)]$. Learn everything from how to sign up for free to enterprise endobj <> \newcommand{\st}{:} stream Math/CS cheat sheet. For complete graph the no . No. 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 of one to one function = (n, P, m)3. WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. /Parent 22 0 R /AIS false ~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. = 6$. Minimum no. /Subtype /Image For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. of functions from A to B = nm2. Combinatorics 71 5.3. Let s = q + r and s = e f be written in lowest terms. 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. From there, he can either choose 4 bus routes or 5 train routes to reach Z. So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. Notes on Discrete Mathematics Representations of Graphs 88 7.3. WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. endobj xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d Toomey.org Tutoring Resources 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 Combinatorial Proofs 1.5 Stars and Bars 1.6 Advanced Counting Using PIE Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? % WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . in the word 'READER'. Bnis the set of binary strings with n bits. Let G be a connected planar simple graph with n vertices and m edges, and no triangles. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. #p Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. endobj 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. >> 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. By using this website, you agree with our Cookies Policy. Did you make this project? ]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 WebDiscrete Math Review n What you should know about discrete math before the midterm. Necessary condition for bijective function |A| = |B|5. Every element has exactly one complement.19. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. WebDefinitions. of asymmetric relations = 3n(n-1)/211. \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. Problem 2 In how many ways can the letters of the word 'READER' be arranged? >> ]\}$ be such that for all $i$, $A_i\neq\varnothing$. If we consider two tasks A and B which are disjoint (i.e. Bipartite Graph : There is no edges between any two vertices of same partition . /Contents 25 0 R o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X 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. stream 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 $S = \lbrace 1, 2, 3 \rbrace$. xKs6. Pascal's identity, first derived by Blaise Pascal in 17th century, states that the number of ways to choose k elements from n elements is equal to the summation of number of ways to choose (k-1) elements from (n-1) elements and the number of ways to choose elements from n-1 elements. Hi matt392, nice work! /Type /XObject 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. /CA 1.0 This number is also called a binomial coefficient since it occurs as a coefficient in the expansion of powers of binomial expressions.Let and be variables and be a non-negative integer. of symmetric relations = 2n(n+1)/29. How many different 10 lettered PAN numbers can be generated such that the first five letters are capital alphabets, the next four are digits and the last is again a capital letter. stream >> stream One of the first things you learn in mathematics is how to count. ]$, 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 = $^np_{n}/n$. Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! xY8_1ow>;|D@`a%e9l96=u=uQ Cartesian ProductsLet A and B be two sets. Equal setsTwo sets are said to be equal if both have same elements. Get up and running with ChatGPT with this comprehensive cheat sheet. Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 Reference Sheet for Discrete Maths - GitHub Pages endobj Cardinality of power set is , where n is the number of elements in a set. How many ways can you choose 3 distinct groups of 3 students from total 9 students? The function is surjective (onto) if every element of the codomain is mapped to by at least one element. Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. / [(a_1!(a_2!) How many ways are there to go from X to Z? 9 years ago Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. of bijection function =n!6. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. element of the domain. + \frac{ n-k } { k!(n-k)! } this looks promising :), Reply Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). on Introduction. Paths and Circuits 91 3 Basic Principles 69 5.2. ];_. \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} >> endobj /ca 1.0 Learn more. Discrete Mathematics 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 . >> endobj Last Minute Notes Discrete Mathematics - GeeksforGeeks