/First 812 Counting Principles - Counting and Cardinality By using our site, you Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, /Type /Page 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. >> (1!)(1!)(2!)] (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! A relation is an equivalence if, 1. Different three digit numbers will be formed when we arrange the digits. I hate discrete math because its hard for me to understand. of symmetric relations = 2n(n+1)/29. *3-d[\HxSi9KpOOHNn uiKa, For two sets A and B, the principle states , $|A \cup B| = |A| + |B| - |A \cap B|$, For three sets A, B and C, the principle states , $|A \cup B \cup C | = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C |$, $|\bigcup_{i=1}^{n}A_i|=\sum\limits_{1\leq iDiscrete Math Cram Sheet - Ateneo de Manila University We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. #p Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K >> endobj xWn7Wgv There must be at least two people in a class of 30 whose names start with the same alphabet. How many ways are there to go from X to Z? Above Venn Diagram shows that A is a subset of B. (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. No. Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? mathematics Share it with us! Prove the following using a proof by contrapositive: Let x be a rational number. 14 0 obj U denotes the universal set. That is, an event is a set consisting of possible outcomes of the experiment. = 180.$. Mathematics | Combinatorics Basics ]\}$ be such that for all $i$, $A_i\neq\varnothing$. >> Learn everything from how to sign up for free to enterprise Then m 3n 6. /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 . There are $50/3 = 16$ numbers which are multiples of 3. }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], 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{\imp}{\rightarrow} \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe 9Wr\Gux[Eul\? 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. /Width 156 Problem 1 From a bunch of 6 different cards, how many ways we can permute it? Combinatorics 71 5.3. This ordered or stable list of counting words must be at least as long as the number of items to be counted. 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. Probability Cheatsheet v2.0 Thinking Conditionally Law of 23 0 obj << In other words a Permutation is an ordered Combination of elements. Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. \newcommand{\N}{\mathbb N} [/Pattern /DeviceRGB] Counting 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. Distributive Lattice : Every Element has zero or 1 complement .18. Course Hero is not sponsored or endorsed by any college or university. Tree, 10. Download the PDF version here. Discrete Math Cheat Sheet by Dois - Cheatography \newcommand{\Iff}{\Leftrightarrow} WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. /ProcSet [ /PDF /Text ] How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. How many integers from 1 to 50 are multiples of 2 or 3 but not both? Probability 78 6.1. 6 0 obj Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional From his home X he has to first reach Y and then Y to Z. 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 $(a_1 + a_2 + a_r) = n$. Extended form of Bayes' rule Let $\{A_i, i\in[\![1,n]\! %PDF-1.2 of connected components in graph with n vertices = n5. Discrete Math 1: Set Theory. Cheat Sheet | by Alex Roan - Medium )$. }$, $= (n-1)! /MediaBox [0 0 612 792] $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. <> \newcommand{\R}{\mathbb R} \newcommand{\inv}{^{-1}} = 720$. 5 0 obj /Parent 22 0 R Representations of Graphs 88 7.3. endobj xY8_1ow>;|D@`a%e9l96=u=uQ There are n number of ways to fill up the first place. From a night class at Fordham University, NYC, Fall, 2008. stream For complete graph the no . Then m 2n 4. }$$. \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} WebE(X)=xP(X=x) (for discreteX) x 1 E(X) =xf(x)dx(for continuousX) TheLaw of the Unconscious Statistician (LOTUS)states thatyou can nd the expected value of afunction of a random WebDefinitions. Toomey.org Tutoring Resources 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. 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$. Probability For Dummies Cheat Sheet - dummies WebDiscrete Math Review n What you should know about discrete math before the midterm. Basic rules to master beginner French! 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. We make use of First and third party cookies to improve our user experience. \newcommand{\Imp}{\Rightarrow} Hence, there are 10 students who like both tea and coffee. ("#} &. a b. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. For solving these problems, mathematical theory of counting are used. Note that zero is an even number, so a string. /AIS false Bipartite Graph : There is no edges between any two vertices of same partition . Maximum no. /SMask /None>> Math ChatGPT cheat sheet: Complete guide for 2023 Web2362 Education Cheat Sheets. WebCheat Sheet of Mathemtical Notation and Terminology Logic and Sets Notation Terminology Explanation and Examples a:=b dened by The objectaon the side of the colon is dened byb. %PDF-1.4 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. The permutation will be = 123, 132, 213, 231, 312, 321, The number of permutations of n different things taken r at a time is denoted by $n_{P_{r}}$. 6 0 obj Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). The Pigeonhole Principle 77 Chapter 6. /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 >> Here's how they described it: Equations commonly used in Discrete Math. 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 Reference Sheet for Discrete Maths - GitHub Pages $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. >> Prove that if xy is irrational, then y is irrational. 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]$. 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 /SM 0.02 By using this website, you agree with our Cookies Policy. Get up and running with ChatGPT with this comprehensive cheat sheet. stream 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. We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. Thus, n2 is odd. Every element has exactly one complement.19. Ten men are in a room and they are taking part in handshakes. \newcommand{\st}{:} + \frac{ n-k } { k!(n-k)! } For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? /ProcSet [ /PDF ] Once we can count, we can determine the likelihood of a particular even and we can estimate how long a A country has two political parties, the Demonstrators and the Repudiators. 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. SA+9)UI)bwKJGJ-4D tFX9LQ Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } Generalized Permutations and Combinations 73 5.4. +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. c o m) stream { r!(n-r)! Examples:x:= 5means thatxis dened to be5, orf.x/ :=x2 *1means that the functionf is dened to bex2 * 1, orA:= ^1;5;7means that the setAis dened to of edges in a complete graph = n(n-1)/22. n Less theory, more problem solving, focuses on exam problems, use as study sheet! { (k-1)!(n-k)! } Cheatsheet - Summary Discrete Mathematics I No. What helped me was to take small bits of information and write them out 25 times or so. I'll check out your sheet when I get to my computer. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! Simple is harder to achieve. ~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. Graph Theory 82 7.1. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. \renewcommand{\iff}{\leftrightarrow} 8"NE!OI6%pu=s{ZW"c"(E89/48q 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. endobj Probability Cheatsheet v1.1.1 Simpsons Paradox Expected CME 106 - Probability Cheatsheet - Stanford University DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream { k!(n-k-1)! 5 0 obj Web445 Cheatsheet. ]\}$ be a partition of the sample space. From there, he can either choose 4 bus routes or 5 train routes to reach Z. Proof Let there be n different elements. The cardinality of the set is 6 and we have to choose 3 elements from the set. | x | = { x if x 0 x if x < 0. /Length 1235 Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. '1g[bXlF) q^|W*BmHYGd tK5A+(R%9;P@2[P9?j28C=r[%\%U08$@`TaqlfEYCfj8Zx!`,O%L v+ ]F$Dx U. Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. /CA 1.0 28 0 obj << /Filter /FlateDecode 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. Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. Power SetsThe power set is the set all possible subset of the set S. Denoted by P(S).Example: What is the power set of {0, 1, 2}?Solution: All possible subsets{}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}.Note: Empty set and set itself is also the member of this set of subsets. You can use all your notes, calcu-lator, and any books you 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. Axioms of probability For each event $E$, we denote $P(E)$ as the probability of event $E$ occurring. of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . Probability 78 Chapter 7. xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d Problem 2 In how many ways can the letters of the word 'READER' be arranged? /Subtype /Image of bijection function =n!6. Notes on Discrete Mathematics There are 6 men and 5 women in a room. WebLet an = rn and substitute for all a terms to get Dividing through by rn2 to get Now we solve this polynomial using the quadratic equation Solve for r to obtain the two roots 1, 2 which is the same as A A +4 B 2 2 r= o If they are distinct, then we get o If they are the same, then we get Now apply initial conditions Graph Theory Types of Graphs 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 Heres something called a theoretical computer science cheat sheet. 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. WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. Bnis the set of binary strings with n bits. on Introduction. CPS102 DISCRETE MATHEMATICS Practice Final Exam - Duke The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. Counting 69 5.1. \newcommand{\isom}{\cong} 9 years ago of relations =2mn7. % Hence, the number of subsets will be $^6C_{3} = 20$. Discrete mathematics cheat sheet Cartesian ProductsLet A and B be two sets. \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} 9 years ago A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. Boolean Lattice: It should be both complemented and distributive. 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. For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. The permutation will be $= 6! Learn more. /Contents 3 0 R + \frac{ (n-1)! } Size of the set S is known as Cardinality number, denoted as |S|. No. 3 0 obj << \[\boxed{P\left(\bigcup_{i=1}^nE_i\right)=\sum_{i=1}^nP(E_i)}\], \[\boxed{C(n, r)=\frac{P(n, r)}{r!}=\frac{n!}{r!(n-r)! 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. Proof : Assume that m and n are both squares. Paths and Circuits 91 3 <> of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. 2195 Last Minute Notes Discrete Mathematics - GeeksforGeeks CS160 - Fall Semester 2015. 17 0 obj 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. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke on April 20, 2023, 5:30 PM EDT. It includes the enumeration or counting of objects having certain properties. Before tackling questions like these, let's look at the basics of counting. No. Discrete Math Cheat Sheet by Dois #education #mathematics #math << of asymmetric relations = 3n(n-1)/211. 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. WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. WebI COUNTING Counting things is a central problem in Discrete Mathematics. It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. By noting $f$ and $F$ the PDF and CDF respectively, we have the following relations: In 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. One of the first things you learn in mathematics is how to count. So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. Then(a+b)modm= ((amodm) + 1 0 obj /CreationDate (D:20151115165753Z) Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide \newcommand{\vl}[1]{\vtx{left}{#1}} Discrete Mathematics - Counting Theory. 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. For solving these problems, mathematical theory of counting are used. Counting mainly encompasses fundamental counting rule, &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 cheat sheet If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. Discrete Mathematics - Counting Theory - TutorialsPoint Let s = q + r and s = e f be written in lowest terms. Basic Principles 69 5.2. If we consider two tasks A and B which are disjoint (i.e. element of the domain. \newcommand{\Z}{\mathbb Z} /Length 58 NOTE: Order of elements of a set doesnt matter. Discrete Mathematics WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. How many ways can you choose 3 distinct groups of 3 students from total 9 students? 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