discrete math counting cheat sheet

\newcommand{\isom}{\cong} Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. 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 $\rho_{XY}\in[-1,1]$. >> Probability 78 6.1. element of the domain. WebBefore tackling questions like these, let's look at the basics of counting. WebDiscrete Math Cram Sheet alltootechnical.tk 7.2 Binomial Coefcients The binomial coefcient (n k) can be dened as the co-efcient of the xk term in the polynomial English to French cheat sheet, with useful words and phrases to take with you on holiday. Before tackling questions like these, let's look at the basics of counting. I'll check out your sheet when I get to my computer. Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). Here it means the absolute value of x, ie. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <2B@>8JfWkn%;?t!yb C;.AIyir!zZn}Na;$t"2b {HEx}]Zg;'B!e>3B=DWw,qS9\ THi_WI04$-1cb I strongly believe that simple is better than complex. 3 0 obj CME 106 - Probability Cheatsheet - Stanford University Discrete Mathematics Cheat Sheet /Type /ExtGState No. In general, use the form The function is injective (one-to-one) if every element of the codomain is mapped to by at most one. \newcommand{\Q}{\mathbb Q} Set DifferenceDifference between sets is denoted by A B, is the set containing elements of set A but not in B. i.e all elements of A except the element of B.ComplementThe complement of a set A, denoted by , is the set of all the elements except A. Complement of the set A is U A. GroupA non-empty set G, (G, *) is called a group if it follows the following axiom: |A| = m and |B| = n, then1. + \frac{ n-k } { k!(n-k)! } Discrete Math Cheat Sheet by Dois #education #mathematics #math Paths and Circuits 91 3 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 $k, \sigma>0$, 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. endobj There are $50/6 = 8$ numbers which are multiples of both 2 and 3. Prove or disprove the following two statements. No. 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. { k!(n-k-1)! 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. WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. endobj of Anti Symmetric Relations = 2n*3n(n-1)/210. Counting Principles - Counting and Cardinality <> << No. Note that zero is an even number, so a string. /SMask /None>> 8"NE!OI6%pu=s{ZW"c"(E89/48q WebIn 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. Hence, there are 10 students who like both tea and coffee. 5 0 obj << Discrete Math Cheat Sheet by Dois - Cheatography \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} 5 0 obj 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! Math/CS cheat sheet. Hence, a+c b+d(modm)andac bd(modm). $A \cap B = \emptyset$), 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. Minimum no. Discrete Mathematics Equal setsTwo sets are said to be equal if both have same elements. Agree Solution There are 6 letters word (2 E, 1 A, 1D and 2R.) What helped me was to take small bits of information and write them out 25 times or so. 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. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. 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} Now, it is known as the pigeonhole principle. /Contents 25 0 R 445 Cheatsheet - Princeton University \definecolor{fillinmathshade}{gray}{0.9} /Type /Page *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! on Introduction. /Parent 22 0 R + \frac{ (n-1)! } /N 100 Get up and running with ChatGPT with this comprehensive cheat sheet. >> 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. %PDF-1.4 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. ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. #p Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K Proof Let there be n different elements. \newcommand{\amp}{&} Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. 23 0 obj << In other words a Permutation is an ordered Combination of elements. of connected components in graph with n vertices = n5. WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. %PDF-1.5 stream \newcommand{\vl}[1]{\vtx{left}{#1}} The no. /Filter /FlateDecode >> How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. Pascal's Identity. In this case the sign means that a divides b, or that b a is an integer. A country has two political parties, the Demonstrators and the Repudiators. \newcommand{\U}{\mathcal U} of edges to have connected graph with n vertices = n-17. /Font << /F17 6 0 R /F18 9 0 R /F15 12 0 R /F7 15 0 R /F8 18 0 R /F37 21 0 R >> 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 $P(a < X\leqslant B)=F(b)-F(a)$. 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. Corollary Let m be a positive integer and let a and b be integers. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } To guarantee that a graph with n vertices is connected, minimum no. Problem 1 From a bunch of 6 different cards, how many ways we can permute it? of relations =2mn7. Sum of degree of all vertices is equal to twice the number of edges.4. The permutation will be $= 6! \renewcommand{\bar}{\overline} Cardinality of power set is , where n is the number of elements in a set. How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? xKs6. Now we want to count large collections of things quickly and precisely. Assume that s is not 0. Discrete Math Cram Sheet - Ateneo de Manila University Combination: A combination of a set of distinct objects is just a count of the number of ways a specific number of elements can be selected from a set of a certain size. Get up and running with ChatGPT with this comprehensive cheat sheet. You can use all your notes, calcu-lator, and any books you on April 20, 2023, 5:30 PM EDT. /Creator () >> endobj CS160 - Fall Semester 2015. In how many ways we can choose 3 men and 2 women from the room? 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. Then, number of permutations of these n objects is = $n! No. x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] Pigeonhole Principle states that if there are fewer pigeon holes than total number of pigeons and each pigeon is put in a pigeon hole, then there must be at least one pigeon hole with more than one pigeon. 592 I dont know whether I agree with the name, but its a nice cheat sheet. >> xm=j0 gRR*9BGRGF. 3 and m edges. Event Any subset $E$ of the sample space is known as an event. xY8_1ow>;|D@`a%e9l96=u=uQ \newcommand{\st}{:} 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. WebChapter 5. /CA 1.0 Math \newcommand{\lt}{<} Then m 3n 6. \newcommand{\va}[1]{\vtx{above}{#1}} of reflexive relations =2n(n-1)8. Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$. /Width 156 endobj endobj /Type /Page ~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. 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 $S = \lbrace 1, 2, 3 \rbrace$. 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.

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