discrete math counting cheat sheet

%PDF-1.3 Cheat Sheet of Mathemtical Notation and Terminology Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. Discrete Mathematics I hate discrete math because its hard for me to understand. /ProcSet [ /PDF ] of spanning tree possible = nn-2. of the domain. \newcommand{\imp}{\rightarrow} cheat sheet of asymmetric relations = 3n(n-1)/211. >> Discrete Mathematics A permutation is an arrangement of some elements in which order matters. /Parent 22 0 R FWfSE xpwy8+3o /MediaBox [0 0 612 792] /Type /ObjStm 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 \renewcommand{\v}{\vtx{above}{}} \renewcommand{\bar}{\overline} 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, \newcommand{\st}{:} WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . The function is surjective (onto) if every element of the codomain is mapped to by at least one element. By using this website, you agree with our Cookies Policy. Proof : Assume that m and n are both squares. \newcommand{\gt}{>} of edges in a complete graph = n(n-1)/22. Maximum no. That is, an event is a set consisting of possible outcomes of the experiment. \newcommand{\inv}{^{-1}} No. From there, he can either choose 4 bus routes or 5 train routes to reach Z. \newcommand{\Imp}{\Rightarrow} Affordable solution to train a team and make them project ready. % So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. Probability For Dummies Cheat Sheet - dummies Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. >> endobj Heres something called a theoretical computer science cheat sheet. Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. /Contents 25 0 R 24 0 obj << Web2362 Education Cheat Sheets. We can also write N+= {x N : x > 0}. WebBefore tackling questions like these, let's look at the basics of counting. \newcommand{\amp}{&} How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. One of the first things you learn in mathematics is how to count. 1 0 obj % Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. Let G be a connected planar simple graph with n vertices and m edges, and no triangles. /Length 7 0 R \newcommand{\R}{\mathbb R} = 720$. on Introduction. /ImageMask true By using our site, you }$, $= (n-1)! on April 20, 2023, 5:30 PM EDT. SA+9)UI)bwKJGJ-4D tFX9LQ 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. endobj Hence, there are (n-2) ways to fill up the third place. Problem 2 In how many ways can the letters of the word 'READER' be arranged? Hi matt392, nice work! How many anagrams are there of anagram? 445 Cheatsheet - Princeton University Hence, there are (n-1) ways to fill up the second place. Ten men are in a room and they are taking part in handshakes. 2195 \newcommand{\N}{\mathbb N} The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. Cartesian ProductsLet A and B be two sets. Webdiscrete math counting cheat sheet.pdf - | Course Hero University of California, Los Angeles MATH MATH 61 discrete math counting cheat sheet.pdf - discrete math Basic Principles 69 5.2. The no. Partition Let $\{A_i, i\in[\![1,n]\! The cardinality of the set is 6 and we have to choose 3 elements from the set. To guarantee that a graph with n vertices is connected, minimum no. /Filter /FlateDecode DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke Boolean Lattice: It should be both complemented and distributive. Prove the following using a proof by contrapositive: Let x be a rational number. }28U*~5} Kryi1#8VVN]dVOJGl\+rlN|~x lsxLw:j(b"&3X]>*~RrKa! Assume that s is not 0. 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID =HA`miNDKH;8&.2_LcVfgsIVAxx$A,t([k9QR$jmOX#Q=s'0z>SUxH-5OPuVq+"a;F} }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. Probability Cheatsheet v1.1.1 Simpsons Paradox Expected Cheat Sheet of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . 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! WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. Probability Cheatsheet v2.0 Thinking Conditionally Law of 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. 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. Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. 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 $P(a < X\leqslant B)=F(b)-F(a)$. 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? /AIS false Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. endobj 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 . stream Size of the set S is known as Cardinality number, denoted as |S|. We say that $\{A_i\}$ is a partition if we have: Remark: for any event $B$ in the sample space, we have $\displaystyle P(B)=\sum_{i=1}^nP(B|A_i)P(A_i)$. Necessary condition for bijective function |A| = |B|5. No. Counting - Discrete Mathematics (c) Express P(k + 1). /MediaBox [0 0 612 792] Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$. \newcommand{\B}{\mathbf B} of irreflexive relations = 2n(n-1), 15. 6 0 obj Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). Math/CS cheat sheet. Discrete Mathematics Cheat Sheet In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. Solution There are 6 letters word (2 E, 1 A, 1D and 2R.) Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). n Less theory, more problem solving, focuses on exam problems, use as study sheet! of functions from A to B = nm2. 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 $n_{P_{r}}$. For complete graph the no . Discrete Math Review $A \cap B = \emptyset$), 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. From a night class at Fordham University, NYC, Fall, 2008. Did you make this project? Definitions // Set A contains elements 1,2 and 3 A = {1,2,3} Prove or disprove the following two statements. How many ways are there to go from X to Z? \newcommand{\vb}[1]{\vtx{below}{#1}} 28 0 obj << ("#} &. DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream 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. Get up and running with ChatGPT with this comprehensive cheat sheet. Helps to encode it into the brain. xY8_1ow>;|D@`a%e9l96=u=uQ &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 on April 20, 2023, 5:30 PM EDT. CPS102 DISCRETE MATHEMATICS Practice Final Exam - Duke Get up and running with ChatGPT with this comprehensive cheat sheet. 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. 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. There are n number of ways to fill up the first place. | x | = { x if x 0 x if x < 0. \newcommand{\C}{\mathbb C} In general, use the form 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. \newcommand{\Iff}{\Leftrightarrow} | x |. 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. Toomey.org Tutoring Resources No. element of the domain. on Introduction. a b. Sum of degree of all vertices is equal to twice the number of edges.4. endobj xKs6. Notes on Discrete Mathematics There are $50/6 = 8$ numbers which are multiples of both 2 and 3. Then m 3n 6. 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 /CA 1.0 endobj (\frac{ k } { k!(n-k)! } Basic rules to master beginner French! { k!(n-k-1)! The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } << @>%c0xC8a%k,s;b !AID/~ Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! Get up and running with ChatGPT with this comprehensive cheat sheet. Every element has exactly one complement.19. 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 $S = \lbrace 1, 2, 3 \rbrace$. How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? 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. Now, it is known as the pigeonhole principle. Binomial Coecients 75 5.5. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 Learn everything from how to sign up for free to enterprise stream /CreationDate (D:20151115165753Z) Representations of Graphs 88 7.3. }$$. 5 0 obj Minimum number of connected components =, 6. stream /Creator () of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. These are my notes created after giving the same lesson 4-5 times in one week. Discrete Math Cheat Sheet by Dois - Cheatography %PDF-1.2 I go out of my way to simplify subjects. /Contents 3 0 R Generalized Permutations and Combinations 73 5.4. WebProof : Assume that n is an odd integer. 3 0 obj <> 17 0 obj (b) Express P(k). stream >> endobj Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. \newcommand{\Q}{\mathbb Q} No. A country has two political parties, the Demonstrators and the Repudiators. of relations =2mn7. WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. )$. Education Cheat Sheets The Pigeonhole Principle 77 Chapter 6. this looks promising :), Reply gQVmDYm*% QKP^n,D%7DBZW=pvh#(sG >> endobj This implies that there is some integer k such that n = 2k + 1. Corollary Let m be a positive integer and let a and b be integers.

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