discrete math counting cheat sheet

Partition Let $\{A_i, i\in[\![1,n]\! 3 0 obj %PDF-1.4 )$. Discrete Mathematics /SMask /None>> ];_. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! Ten men are in a room and they are taking part in handshakes. /Parent 22 0 R on April 20, 2023, 5:30 PM EDT. Hence, a+c b+d(modm)andac bd(modm). 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 5 0 obj << \newcommand{\va}[1]{\vtx{above}{#1}} That's a good collection you've got there, but your typesetting is aweful, I could help you with that. x3T0 BCKs=S\.t;!THcYYX endstream of edges =m*n3. @>%c0xC8a%k,s;b !AID/~ }$$. I go out of my way to simplify subjects. >> Corollary Let m be a positive integer and let a and b be integers. Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! WebDiscrete Math Review n What you should know about discrete math before the midterm. \(\renewcommand{\d}{\displaystyle} \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} /Type /XObject Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. /Decode [1 0] Binomial Coecients 75 5.5. of relations =2mn7. Assume that s is not 0. *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! /SA true Learn everything from how to sign up for free to enterprise endobj 9 years ago xWn7Wgv WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. cheat sheet stream The no. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream >> endobj Pascal's Identity. Probability 78 6.1. \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} Vertical bar sign in Discrete mathematics \newcommand{\lt}{<} %PDF-1.5 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} \newcommand{\Imp}{\Rightarrow} 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 From a night class at Fordham University, NYC, Fall, 2008. % >> endobj &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 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. 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. Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). \newcommand{\card}[1]{\left| #1 \right|} @ys(5u$E$VY(@[Y+J(or(0ze7+s([nlY+J(or(0zemFGn2+%f mEH(X $c62MC*u+Z The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. 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)$. on Introduction. [/Pattern /DeviceRGB] [Q hm*q*E9urWYN#-&\" e1cU3D).C5Q7p66[XlG|;xvvANUr_B(mVt2pzbShb5[Tv!k":,7a) 25 0 obj << I'll check out your sheet when I get to my computer. + \frac{ (n-1)! } Discrete Mathematics - Counting Theory - TutorialsPoint \renewcommand{\v}{\vtx{above}{}} 8"NE!OI6%pu=s{ZW"c"(E89/48q 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$. 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 /Length 7 0 R \newcommand{\st}{:} How many integers from 1 to 50 are multiples of 2 or 3 but not both? Graphs 82 7.2. WebBefore tackling questions like these, let's look at the basics of counting. 2 0 obj << /MediaBox [0 0 612 792] \newcommand{\vl}[1]{\vtx{left}{#1}} Affordable solution to train a team and make them project ready. Paths and Circuits 91 3 This implies that there is some integer k such that n = 2k + 1. /N 100 Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). \newcommand{\N}{\mathbb N} /Type /ObjStm /Length 530 *3-d[\HxSi9KpOOHNn uiKa, 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. ("#} &. Notes on Discrete Mathematics 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. Tree, 10. /AIS false 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 a b. WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <= n; 0 otherwise4. 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. 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. 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)$. Hence, the number of subsets will be $^6C_{3} = 20$. /ca 1.0 Then(a+b)modm= ((amodm) + Cardinality of power set is , where n is the number of elements in a set. Harold's Cheat Sheets "If you can't explain it simply, you don't understand it well enough." BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. of edges required = {(n-1)*(n-2)/2 } + 18. 17 0 obj A relation is an equivalence if, 1. Thank you - hope it helps. 4 0 obj Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. Necessary condition for bijective function |A| = |B|5. Discrete Mathematics Cheat Sheet - DocDroid /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 >> In complete bipartite graph no. 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. 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! \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} Let s = q + r and s = e f be written in lowest terms. xY8_1ow>;|D@`a%e9l96=u=uQ mathematics 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. / [(a_1!(a_2!) >> endobj For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. | x | = { x if x 0 x if x < 0. endobj Let G be a connected planar simple graph with n vertices and m edges, and no triangles. Then m 2n 4. discrete math counting cheat sheet.pdf - | Course Hero Discrete Math Cram Sheet - Ateneo de Manila University No. /Resources 23 0 R 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! Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). Toomey.org Tutoring Resources We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. of symmetric relations = 2n(n+1)/29. >> Maximum no. Event Any subset $E$ of the sample space is known as an event. \dots (a_r!)]$. Note that zero is an even number, so a string. Here it means the absolute value of x, ie. 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. 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. Then, The binomial expansion using Combinatorial symbols. `y98R uA>?2 AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. What helped me was to take small bits of information and write them out 25 times or so. 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. Complemented Lattice : Every element has complement17. of reflexive relations =2n(n-1)8. \newcommand{\R}{\mathbb R} So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d Proof Let there be n different elements. \newcommand{\imp}{\rightarrow} Let G be a connected planar simple graph with n vertices, where n ? WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. Representations of Graphs 88 7.3. Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. 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. Counting Simple is harder to achieve. Cheatsheet - Summary Discrete Mathematics I Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. Solution There are 6 letters word (2 E, 1 A, 1D and 2R.) 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 $w_1 + w_2 + \dots +w_m$. Question A boy lives at X and wants to go to School at Z. Generalized Permutations and Combinations 73 5.4. I dont know whether I agree with the name, but its a nice cheat sheet. From his home X he has to first reach Y and then Y to Z. Boolean Lattice: It should be both complemented and distributive. x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] Probability Cheatsheet v2.0 Thinking Conditionally Law of SA+9)UI)bwKJGJ-4D tFX9LQ Prove that if xy is irrational, then y is irrational. $A \cap B = \emptyset$), 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.

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