discrete math counting cheat sheet

/ProcSet [ /PDF ] 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. \newcommand{\C}{\mathbb C} The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! \newcommand{\imp}{\rightarrow} /ca 1.0 /CA 1.0 Size of a SetSize of a set can be finite or infinite. WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. Definitions // Set A contains elements 1,2 and 3 A = {1,2,3} Graph Theory 82 7.1. >> endobj BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. Problem 1 From a bunch of 6 different cards, how many ways we can permute it? | x | = { x if x 0 x if x < 0. Cheatsheet - Summary Discrete Mathematics I There are two very important equivalences involving quantifiers. That ("#} &. Hence, the number of subsets will be $^6C_{3} = 20$. There are n number of ways to fill up the first place. }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. >> Discrete Structures Lecture Notes - Stanford University on April 20, 2023, 5:30 PM EDT. on April 20, 2023, 5:30 PM EDT. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! /Filter /FlateDecode Once we can count, we can determine the likelihood of a particular even and we can estimate how long a computer algorithm takes to complete a task. endobj WebBefore tackling questions like these, let's look at the basics of counting. In other words a Permutation is an ordered Combination of elements. \newcommand{\gt}{>} xS@}WD"f<7.\$.iH(Rc'vbo*g1@9@I4_ F2 }3^C2>2B@>8JfWkn%;?t!yb C;.AIyir!zZn}Na;$t"2b {HEx}]Zg;'B!e>3B=DWw,qS9\ THi_WI04$-1cb (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. 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. Probability 78 6.1. 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 Cheat Sheet by Dois #education #mathematics #math 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. Discrete Mathematics Cheat Sheet << endobj Basic Principles 69 5.2. How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke Event Any subset $E$ of the sample space is known as an event. Discrete mathematics cheat sheet Size of the set S is known as Cardinality number, denoted as |S|. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. 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 $P(n,r)\geqslant C(n,r)$. 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. Discrete Math Cheat Sheet by Dois - Cheatography ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream }28U*~5} Kryi1#8VVN]dVOJGl\+rlN|~x lsxLw:j(b"&3X]>*~RrKa! From a night class at Fordham University, NYC, Fall, 2008. :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. No. '1g[bXlF) q^|W*BmHYGd tK5A+(R%9;P@2[P9?j28C=r[%\%U08$@`TaqlfEYCfj8Zx!`,O%L v+ ]F$Dx U. CME 106 - Probability Cheatsheet - Stanford University \newcommand{\Iff}{\Leftrightarrow} of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? Prove the following using a proof by contrapositive: Let x be a rational number. = 6$. 1 0 obj << \definecolor{fillinmathshade}{gray}{0.9} \). Web2362 Education Cheat Sheets. \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} 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. 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 $n_{P_{r}}$. A combination is selection of some given elements in which order does not matter. Once we can count, we can determine the likelihood of a particular even and we can estimate how long a The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. 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 discrete math counting cheat sheet.pdf - | Course Hero on Introduction. Discrete Mathematics I'll check out your sheet when I get to my computer. Webdiscrete math counting cheat sheet.pdf - | Course Hero University of California, Los Angeles MATH MATH 61 discrete math counting cheat sheet.pdf - discrete math We can also write N+= {x N : x > 0}. A poset is called Lattice if it is both meet and join semi-lattice16. 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 . /Length 1781 Here's how they described it: Equations commonly used in Discrete Math. Toomey.org Tutoring Resources Bipartite Graph : There is no edges between any two vertices of same partition . The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. 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. *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! )$. WebThe first principle of counting involves the student using a list of words to count in a repeatable order. ];_. In this case it is written with just the | symbol. 9 years ago \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. >> Hence, the total number of permutation is $6 \times 6 = 36$. Probability Cheatsheet v1.1.1 Simpsons Paradox Expected 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. xWn7Wgv These are my notes created after giving the same lesson 4-5 times in one week. 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 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. of reflexive relations =2n(n-1)8. + \frac{ n-k } { k!(n-k)! } How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. Generalized Permutations and Combinations 73 5.4. How many ways are there to go from X to Z? \newcommand{\amp}{&} SA+9)UI)bwKJGJ-4D tFX9LQ 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. << % /Filter /FlateDecode = 180.$. = 6$ ways. stream Cheat Sheet of Mathemtical Notation and Terminology I strongly believe that simple is better than complex. (c) Express P(k + 1). >> Necessary condition for bijective function |A| = |B|5. \newcommand{\va}[1]{\vtx{above}{#1}} of functions from A to B = nm2. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. 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 Then(a+b)modm= ((amodm) + /Height 25 /\: [(2!) No. WebDiscrete Math Review n What you should know about discrete math before the midterm. 4 0 obj Education Cheat Sheets No. How many ways can you choose 3 distinct groups of 3 students from total 9 students? 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? 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 The no. 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. 24 0 obj << Partition Let $\{A_i, i\in[\![1,n]\! The permutation will be $= 6! Prove or disprove the following two statements. Simple is harder to achieve. stream After filling the first place (n-1) number of elements is left. \newcommand{\isom}{\cong} A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. Discrete Math Cram Sheet - Ateneo de Manila University Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). of edges in a complete graph = n(n-1)/22. /ProcSet [ /PDF /Text ] Mathematics | Combinatorics Basics Learn everything from how to sign up for free to enterprise Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. Maximum no. From his home X he has to first reach Y and then Y to Z. Hi matt392, nice work! 3 and m edges. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. Counting \renewcommand{\bar}{\overline} %PDF-1.2 We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. Notes on Discrete Mathematics IntersectionThe intersection of the sets A and B, denoted by A B, is the set of elements belongs to both A and B i.e. <> /Contents 3 0 R 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. 5 0 obj Get up and running with ChatGPT with this comprehensive cheat sheet. You can use all your notes, calcu-lator, and any books you No. of bijection function =n!6. /SM 0.02 A relation is an equivalence if, 1. }$$. I go out of my way to simplify subjects. + \frac{ (n-1)! } No. Show that if m and n are both square numbers, then m n is also a square number. Let G be a connected planar simple graph with n vertices, where n ? (\frac{ k } { k!(n-k)! } Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 stream Last Minute Notes Discrete Mathematics - GeeksforGeeks Affordable solution to train a team and make them project ready. 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 $(a_1 + a_2 + a_r) = n$. %PDF-1.4 CPS102 DISCRETE MATHEMATICS Practice Final Exam - Duke Boolean Lattice: It should be both complemented and distributive. /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 . \renewcommand{\v}{\vtx{above}{}} /CreationDate (D:20151115165753Z) >> endobj Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$.

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