There are $50/3 = 16$ numbers which are multiples of 3. WebDiscrete Math Review n What you should know about discrete math before the midterm. Binomial Coecients 75 5.5. (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! \dots (a_r!)]$. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. It includes the enumeration or counting of objects having certain properties. Here's how they described it: Equations commonly used in Discrete Math. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. 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. 3 0 obj of connected components in graph with n vertices = n5. Boolean Lattice: It should be both complemented and distributive. No. 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. Every element has exactly one complement.19. = 720$. /Height 25 The cardinality of the set is 6 and we have to choose 3 elements from the set. +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X Event Any subset $E$ of the sample space is known as an event. % 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 $\rho_{XY}\in[-1,1]$. One of the first things you learn in mathematics is how to count. 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} How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? \newcommand{\lt}{<} ~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. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). A poset is called Lattice if it is both meet and join semi-lattice16. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe
9Wr\Gux[Eul\? Before tackling questions like these, let's look at the basics of counting. /Length 1235 We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. 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 of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. The function is injective (one-to-one) if every element of the codomain is mapped to by at most one. \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. Thus, n2 is odd. >> endobj = 180.$. endobj %PDF-1.4 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. Webdiscrete math counting cheat sheet.pdf - | Course Hero University of California, Los Angeles MATH MATH 61 discrete math counting cheat sheet.pdf - discrete math This ordered or stable list of counting words must be at least as long as the number of items to be counted. We make use of First and third party cookies to improve our user experience. endobj /Filter /FlateDecode /CA 1.0 /CreationDate (D:20151115165753Z) Prove the following using a proof by contrapositive: Let x be a rational number. WebDefinitions. Cardinality of power set is , where n is the number of elements in a set. Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. In this case the sign means that a divides b, or that b a is an integer. See Last Minute Notes on all subjects here. /Filter /FlateDecode E(aX+bY+c) =aE(X) +bE(Y) +c If two Random Variables have the same distribution, even when theyare dependent by theproperty of Symmetrytheir expected of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . /N 100 )$. Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. 23 0 obj << / [(a_1!(a_2!) /ProcSet [ /PDF ] /Length 7 0 R (1!)(1!)(2!)] Learn more. 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 Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! Pigeonhole Principle states that if there are fewer pigeon holes than total number of pigeons and each pigeon is put in a pigeon hole, then there must be at least one pigeon hole with more than one pigeon. \newcommand{\Imp}{\Rightarrow} Download the PDF version here. /Type /XObject Get up and running with ChatGPT with this comprehensive cheat sheet. (b) Express P(k). The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. Necessary condition for bijective function |A| = |B|5. The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. 5 0 obj /Filter /FlateDecode this looks promising :), Reply of irreflexive relations = 2n(n-1), 15. From a night class at Fordham University, NYC, Fall, 2008. on Introduction. { k!(n-k-1)! WebDiscrete and Combinatorial Mathematics. Examples:x:= 5means thatxis dened to be5, orf.x/ :=x2 *1means that the functionf is dened to bex2 * 1, orA:= ^1;5;7means that the setAis dened to \newcommand{\Q}{\mathbb Q} ]$, The number of circular permutations of n different elements taken x elements at time = $^np_{x}/x$, The number of circular permutations of n different things = $^np_{n}/n$. /\: [(2!) Extended form of Bayes' rule Let $\{A_i, i\in[\![1,n]\! 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 $(a_1 + a_2 + a_r) = n$. 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 $c$. Equal setsTwo sets are said to be equal if both have same elements. 9 years ago In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. a b. If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. stream ]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 endobj Helps to encode it into the brain. 2195 /Creator () >> Assume that s is not 0. \newcommand{\N}{\mathbb N} Hi matt392, nice work! stream I go out of my way to simplify subjects. No. There are n number of ways to fill up the first place. In this case it is written with just the | symbol. of asymmetric relations = 3n(n-1)/211. WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. stream \renewcommand{\iff}{\leftrightarrow} WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. Axiom 1 Every probability is between 0 and 1 included, i.e: Axiom 2 The probability that at least one of the elementary events in the entire sample space will occur is 1, i.e: Axiom 3 For any sequence of mutually exclusive events $E_1, , E_n$, we have: Permutation A permutation is an arrangement of $r$ objects from a pool of $n$ objects, in a given order. #p
Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K 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. \newcommand{\B}{\mathbf B} of edges to have connected graph with n vertices = n-17. *3-d[\HxSi9KpOOHNn uiKa, 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. SA+9)UI)bwKJGJ-4D
tFX9LQ >> 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 % 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. ];_. /MediaBox [0 0 612 792] Distributive Lattice : Every Element has zero or 1 complement .18. Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. /Contents 25 0 R /ProcSet [ /PDF /Text ] | x | = { x if x 0 x if x < 0. WebThe first principle of counting involves the student using a list of words to count in a repeatable order. Once we can count, we can determine the likelihood of a particular even and we can estimate how long a A permutation is an arrangement of some elements in which order matters. 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. Affordable solution to train a team and make them project ready. 24 0 obj << How many ways are there to go from X to Z? The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } How many ways can you choose 3 distinct groups of 3 students from total 9 students? Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional In complete bipartite graph no. 17 0 obj @ys(5u$E$VY(@[Y+J(or(0ze7+s([nlY+J(or(0zemFGn2+%f mEH(X 3 and m edges. x3T0 BCKs=S\.t;!THcYYX endstream }$$. Equivalesistheonlyequivalencerelationthatisassociative ((p q) r) (p (q WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. 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 $S = \lbrace 1, 2, 3 \rbrace$. /Resources 1 0 R 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 endobj Proof Let there be n different elements. Get up and running with ChatGPT with this comprehensive cheat sheet. \newcommand{\vb}[1]{\vtx{below}{#1}} Then, number of permutations of these n objects is = $n! 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. \). Hence, the number of subsets will be $^6C_{3} = 20$. element of the domain. (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. \(\renewcommand{\d}{\displaystyle} /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 . /SM 0.02 14 0 obj WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. 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. 5 0 obj << ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream /Producer ( w k h t m l t o p d f) Then, The binomial expansion using Combinatorial symbols. That Web445 Cheatsheet. /Parent 22 0 R /Resources 23 0 R WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <} 6 0 obj Generalized Permutations and Combinations 73 5.4. \newcommand{\U}{\mathcal U} stream 1 This is a matter of taste. Representations of Graphs 88 7.3. &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 Web2362 Education Cheat Sheets. These are my notes created after giving the same lesson 4-5 times in one week. A Set is an unordered collection of objects, known as elements or members of the set.An element a belong to a set A can be written as a ∈ A, a A denotes that a is not an element of the set A. 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. Prove or disprove the following two statements. >> If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. 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! Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] \newcommand{\Z}{\mathbb Z} '1g[bXlF) q^|W*BmHYGd tK5A+(R%9;P@2[P9?j28C=r[%\%U08$@`TaqlfEYCfj8Zx!`,O%L v+ ]F$Dx U. ("#} &. No. Complemented Lattice : Every element has complement17. in the word 'READER'. There are two very important equivalences involving quantifiers. For solving these problems, mathematical theory of counting are used. Note that zero is an even number, so a string. For complete graph the no . WebProof : Assume that n is an odd integer. Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. NOTE: Order of elements of a set doesnt matter. 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. Graphs 82 7.2. /Type /ExtGState xWn7Wgv Bipartite Graph : There is no edges between any two vertices of same partition . Maximum no. Hence, the total number of permutation is $6 \times 6 = 36$. \newcommand{\st}{:} >> endobj How many integers from 1 to 50 are multiples of 2 or 3 but not both? Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. 4 0 obj endobj 5 0 obj A combination is selection of some given elements in which order does not matter. How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. There must be at least two people in a class of 30 whose names start with the same alphabet. of spanning tree possible = nn-2. \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} \renewcommand{\bar}{\overline} Pascal's identity, first derived by Blaise Pascal in 17 century, states that 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. Probability 78 6.1. set of the common element in A and B. DisjointTwo sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements. Paths and Circuits 91 3 It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. 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? \newcommand{\C}{\mathbb C} <> $c62MC*u+Z To guarantee that a graph with n vertices is connected, minimum no. 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. No. Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. + \frac{ (n-1)! } of edges in a complete graph = n(n-1)/22. /First 812 on April 20, 2023, 5:30 PM EDT. Then(a+b)modm= ((amodm) + Probability 78 Chapter 7. \newcommand{\va}[1]{\vtx{above}{#1}} Problem 2 In how many ways can the letters of the word 'READER' be arranged? In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. of bijection function =n!6. 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 $P(n,r)\geqslant C(n,r)$. c o m) Hence, a+c b+d(modm)andac bd(modm). How many like both coffee and tea? \newcommand{\vl}[1]{\vtx{left}{#1}} }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$.
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