discrete math counting cheat sheet

Assume that s is not 0. No. From his home X he has to first reach Y and then Y to Z. on Introduction. 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 Hence, the number of subsets will be $^6C_{3} = 20$. \newcommand{\amp}{&} Then, number of permutations of these n objects is = $n! Get up and running with ChatGPT with this comprehensive cheat sheet. Bipartite Graph : There is no edges between any two vertices of same partition . /Width 156 %PDF-1.4 Proof Let there be n different elements. { (k-1)!(n-k)! } a b. /ImageMask true Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. xY8_1ow>;|D@`a%e9l96=u=uQ /SMask /None>> 3 0 obj << Prove or disprove the following two statements. \definecolor{fillinmathshade}{gray}{0.9} Bnis the set of binary strings with n bits. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. #p Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K >> endobj In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. /SA true A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. /CreationDate (D:20151115165753Z) 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 number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. \newcommand{\B}{\mathbf B} Web2362 Education Cheat Sheets. \newcommand{\vr}[1]{\vtx{right}{#1}} There are n number of ways to fill up the first place. stream Here, the ordering does not matter. endobj Minimum no. Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe 9Wr\Gux[Eul\? Basic Principles 69 5.2. Cardinality of power set is , where n is the number of elements in a set. /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 >> stream Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. on Introduction. There are 6 men and 5 women in a room. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. /Subtype /Image Let q = a b and r = c d be two rational numbers written in lowest terms. We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. 1 This is a matter of taste. WebDefinitions. Web445 Cheatsheet. 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 $(a_1 + a_2 + a_r) = n$. Different three digit numbers will be formed when we arrange the digits. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. /Filter /FlateDecode 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. >> = 6$ ways. 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. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. %PDF-1.2 24 0 obj << Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. | x |. It includes the enumeration or counting of objects having certain properties. = 180.$. /Decode [1 0] Sum of degree of all vertices is equal to twice the number of edges.4. Affordable solution to train a team and make them project ready. 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 $c$. This ordered or stable list of counting words must be at least as long as the number of items to be counted. /Filter /FlateDecode Boolean Lattice: It should be both complemented and distributive. /Filter /FlateDecode Learn more. A country has two political parties, the Demonstrators and the Repudiators. This implies that there is some integer k such that n = 2k + 1. How many ways are there to go from X to Z? A poset is called Lattice if it is both meet and join semi-lattice16. I hate discrete math because its hard for me to understand. No. If each person shakes hands at least once and no man shakes the same mans hand more than once then two men took part in the same number of handshakes. Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. 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. Define P(n) to be the assertion that: j=1nj2=n(n+1)(2n+1)6 (a) Verify that P(3) is true. 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 $S = \lbrace 1, 2, 3 \rbrace$. WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. 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. 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! /Contents 25 0 R Heres something called a theoretical computer science cheat sheet. << Let G be a connected planar simple graph with n vertices and m edges, and no triangles. \newcommand{\va}[1]{\vtx{above}{#1}} stream stream \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. /Length 530 4 0 obj 1 0 obj << Show that if m and n are both square numbers, then m n is also a square number. 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. Note that zero is an even number, so a string. /AIS false of edges required = {(n-1)*(n-2)/2 } + 18. <> 3 and m edges. 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. \[\boxed{P\left(\bigcup_{i=1}^nE_i\right)=\sum_{i=1}^nP(E_i)}\], \[\boxed{C(n, r)=\frac{P(n, r)}{r!}=\frac{n!}{r!(n-r)! Math/CS cheat sheet. 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. For solving these problems, mathematical theory of counting are used. &IP")0 QlaK5 )CPq 9n TVd,L j' )3 O@ 3+$ >+:>Ov?! How many anagrams are there of anagram? 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. That of the domain. [/Pattern /DeviceRGB] WebProof : Assume that n is an odd integer. ("#} &. Probability 78 Chapter 7. }28U*~5} Kryi1#8VVN]dVOJGl\+rlN|~x lsxLw:j(b"&3X]>*~RrKa! ]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 592 /CA 1.0 >> WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. After filling the first and second place, (n-2) number of elements is left. WebChapter 5. Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. [Q hm*q*E9urWYN#-&\" e1cU3D).C5Q7p66[XlG|;xvvANUr_B(mVt2pzbShb5[Tv!k":,7a) >> 23 0 obj << /Contents 3 0 R Then m 3n 6. Every element has exactly one complement.19. 6 0 obj 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. By using our site, you

Tjx Distribution Center Locations, Bungalows To Rent In Sileby, Misconceptions With The Key Objectives Ncetm, Amber Digiovanni Kansas City Home, 22 Week Infantry Osut Schedule, Articles D