Download the PDF version here. 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. 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. How many integers from 1 to 50 are multiples of 2 or 3 but not both? xmT;s1Wli+,[-:^Q1GL$E=>]KC}{~=ogwh=9-} }pNY@z }>c? Pascal's Identity. /Type /XObject 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} of edges to have connected graph with n vertices = n-17. { r!(n-r)! WebThe first principle of counting involves the student using a list of words to count in a repeatable order. &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 Paths and Circuits 91 3 Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. /ca 1.0 { k!(n-k-1)! If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. of symmetric relations = 2n(n+1)/29. /CA 1.0 Problem 2 In how many ways can the letters of the word 'READER' be arranged? Axioms of probability For each event $E$, we denote $P(E)$ as the probability of event $E$ occurring. /Type /ExtGState The Pigeonhole Principle 77 Chapter 6. /Parent 22 0 R /Parent 22 0 R 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. 4 0 obj Corollary Let m be a positive integer and let a and b be integers. I dont know whether I agree with the name, but its a nice cheat sheet. There are two very important equivalences involving quantifiers. %PDF-1.3 *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! \(\renewcommand{\d}{\displaystyle} stream of Anti Symmetric Relations = 2n*3n(n-1)/210. Size of a SetSize of a set can be finite or infinite. After filling the first place (n-1) number of elements is left. I strongly believe that simple is better than complex. In general, use the form FWfSE
xpwy8+3o 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 $\rho_{XY}\in[-1,1]$. >> Then(a+b)modm= ((amodm) + }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. \renewcommand{\iff}{\leftrightarrow} Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. Assume that s is not 0. 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 $c$. What helped me was to take small bits of information and write them out 25 times or so. /Length 58 /MediaBox [0 0 612 792] There are $50/3 = 16$ numbers which are multiples of 3. 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 17 0 obj For complete graph the no . %PDF-1.4 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. << 9 years ago Let G be a connected planar simple graph with n vertices and m edges, and no triangles. See Last Minute Notes on all subjects here. )$. Now, it is known as the pigeonhole principle. /SA true %PDF-1.4 It includes the enumeration or counting of objects having certain properties. xS@}WD"f<7.\$.iH(Rc'vbo*g1@9@I4_ F2
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{HEx}]Zg;'B!e>3B=DWw,qS9\ THi_WI04$-1cb Did you make this project? of edges required = {(n-1)*(n-2)/2 } + 18. Then, The binomial expansion using Combinatorial symbols. Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. stream I hate discrete math because its hard for me to understand. We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. Thus, n2 is odd. ("#} &. It wasn't meant to be a presentation per se, but more of a study sheet, so I did not work too hard on the typesetting. /Decode [1 0] of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . 1 This is a matter of taste. >> endobj element of the domain. (c) Express P(k + 1). /Type /ObjStm Prove the following using a proof by contrapositive: Let x be a rational number. $c62MC*u+Z 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. We can also write N+= {x N : x > 0}. The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! Extended form of Bayes' rule Let $\{A_i, i\in[\![1,n]\! $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. /Length 1235 @ys(5u$E$VY(@[Y+J(or(0ze7+s([nlY+J(or(0zemFGn2+%f mEH(X Get up and running with ChatGPT with this comprehensive cheat sheet. \newcommand{\R}{\mathbb R} These are my notes created after giving the same lesson 4-5 times in one week. Counting 69 5.1. 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. :oCH7ZG_
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?K?*]ZrLbu7,J^(80~*@dL"rjx /\: [(2!) stream The function is surjective (onto) if every element of the codomain is mapped to by at least one element. /ProcSet [ /PDF /Text ] 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 Bipartite Graph : There is no edges between any two vertices of same partition . WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. WebDiscrete Math Review n What you should know about discrete math before the midterm. Now we want to count large collections of things quickly and precisely. /AIS false There are n number of ways to fill up the first place. endobj There are 6 men and 5 women in a room. Graph Theory 82 7.1. Affordable solution to train a team and make them project ready. 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. 2195 Proof Let there be n different elements. It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. Bnis the set of binary strings with n bits. One of the first things you learn in mathematics is how to count. >> If we consider two tasks A and B which are disjoint (i.e. 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 $(a_1 + a_2 + a_r) = n$. >> endobj Hence, a+c b+d(modm)andac bd(modm). Event Any subset $E$ of the sample space is known as an event. /Contents 3 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! \newcommand{\Z}{\mathbb Z} U denotes the universal set. 14 0 obj + \frac{ n-k } { k!(n-k)! } \newcommand{\gt}{>} \renewcommand{\bar}{\overline} By using our site, you of the domain. stream 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. Boolean Lattice: It should be both complemented and distributive. 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 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{\isom}{\cong} of edges in a complete graph = n(n-1)/22. /Filter /FlateDecode Hence, there are (n-2) ways to fill up the third place. ];_. 9 years ago *3-d[\HxSi9KpOOHNn uiKa, I go out of my way to simplify subjects. Then m 3n 6. Hence, there are (n-1) ways to fill up the second place. Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. CS160 - Fall Semester 2015. xY8_1ow>;|D@`a%e9l96=u=uQ 6 0 obj endobj 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. No. \newcommand{\card}[1]{\left| #1 \right|} 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, No. `y98R uA>?2
AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! /Filter /FlateDecode How many ways are there to go from X to Z? stream In this case it is written with just the | symbol. [Q
hm*q*E9urWYN#-&\" e1cU3D).C5Q7p66[XlG|;xvvANUr_B(mVt2pzbShb5[Tv!k":,7a) How many like both coffee and tea? Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } In other words a Permutation is an ordered Combination of elements. 2 0 obj << Graphs 82 7.2. Hence, there are 10 students who like both tea and coffee. x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. endobj 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 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. Once we can count, we can determine the likelihood of a particular even and we can estimate how long a 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists You can use all your notes, calcu-lator, and any books you <> \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. 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. 1 0 obj (1!)(1!)(2!)] Thank you - hope it helps. /SM 0.02 The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. /Filter /FlateDecode /Filter /FlateDecode of reflexive relations =2n(n-1)8. xm=j0 gRR*9BGRGF. 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. /Resources 23 0 R Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. %PDF-1.5 I have a class in it right now actually! If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. ]\}$ be such that for all $i$, $A_i\neq\varnothing$. Here, the ordering does not matter. WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. There are $50/6 = 8$ numbers which are multiples of both 2 and 3. Course Hero is not sponsored or endorsed by any college or university. \newcommand{\B}{\mathbf B} of bijection function =n!6. A relation is an equivalence if, 1. No. Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? on April 20, 2023, 5:30 PM EDT. << So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. 23 0 obj << c o m) 6 0 obj 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. Problem 1 From a bunch of 6 different cards, how many ways we can permute it? There must be at least two people in a class of 30 whose names start with the same alphabet. Cardinality of power set is , where n is the number of elements in a set. /SMask /None>> = 180.$. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. Hence, the number of subsets will be $^6C_{3} = 20$. Partition Let $\{A_i, i\in[\![1,n]\! <> A combination is selection of some given elements in which order does not matter. 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 i> NOTE: Order of elements of a set doesnt matter. | x |. Here's how they described it: Equations commonly used in Discrete Math. stream This ordered or stable list of counting words must be at least as long as the number of items to be counted. Hence, the total number of permutation is $6 \times 6 = 36$. Generalized Permutations and Combinations 73 5.4. o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X WebCounting things is a central problem in Discrete Mathematics. Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. 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. /Type /Page 592 No. Learn everything from how to sign up for free to enterprise No. \newcommand{\vb}[1]{\vtx{below}{#1}} Probability 78 Chapter 7. (nr+1)! Pascal's identity, first derived by Blaise Pascal in 17 century, states that Every element has exactly one complement.19. Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. \newcommand{\imp}{\rightarrow} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} 5 0 obj /Producer ( w k h t m l t o p d f) 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID
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>+:>Ov?! \newcommand{\st}{:} Pascal's identity, first derived by Blaise Pascal in 17th century, states that the number of ways to choose k elements from n elements is equal to the summation of number of ways to choose (k-1) elements from (n-1) elements and the number of ways to choose elements from n-1 elements. on Introduction. Equal setsTwo sets are said to be equal if both have same elements. 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. Prove or disprove the following two statements. Heres something called a theoretical computer science cheat sheet. Prove that if xy is irrational, then y is irrational. 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. 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. WebProof : Assume that n is an odd integer. BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. on April 20, 2023, 5:30 PM EDT. 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. It is determined as follows: Characteristic function A characteristic function $\psi(\omega)$ is derived from a probability density function $f(x)$ and is defined as: Euler's formula For $\theta \in \mathbb{R}$, the Euler formula is the name given to the identity: Revisiting the $k^{th}$ moment The $k^{th}$ moment can also be computed with the characteristic function as follows: Transformation of random variables Let the variables $X$ and $Y$ be linked by some function. Besides, your proof of 0!=1 needs some more attention. on April 20, 2023, 5:30 PM EDT. Different three digit numbers will be formed when we arrange the digits. 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.
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