What helped me was to take small bits of information and write them out 25 times or so. endobj There are two very important equivalences involving quantifiers. >> /Contents 25 0 R Define the set Ento be the set of binary strings with n bits that have an even number of 1's. 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. /Filter /FlateDecode /Resources 1 0 R WebDiscrete and Combinatorial Mathematics. In general, use the form 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. From his home X he has to first reach Y and then Y to Z. The permutation will be $= 6! stream Assume that s is not 0. 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 $P(n,r)\geqslant C(n,r)$. By using our site, you \newcommand{\Q}{\mathbb Q} I dont know whether I agree with the name, but its a nice cheat sheet. / [(a_1!(a_2!) If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. /Creator () 1 0 obj << 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 $c$. Probability 78 Chapter 7. Sum of degree of all vertices is equal to twice the number of edges.4. FWfSE
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Simple is harder to achieve. \newcommand{\U}{\mathcal U} /Type /XObject No. }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. of Anti Symmetric Relations = 2n*3n(n-1)/210. 24 0 obj << { k!(n-k-1)! 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} *3-d[\HxSi9KpOOHNn uiKa, Agree For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. on April 20, 2023, 5:30 PM EDT. %PDF-1.2 %PDF-1.3 $A \cap B = \emptyset$), 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. Heres something called a theoretical computer science cheat sheet. \newcommand{\gt}{>} Hence, a+c b+d(modm)andac bd(modm). 2 0 obj << Hence, there are 10 students who like both tea and coffee. of asymmetric relations = 3n(n-1)/211. I have a class in it right now actually! \newcommand{\imp}{\rightarrow} 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 . \newcommand{\B}{\mathbf B} >> /Producer ( w k h t m l t o p d f) English to French cheat sheet, with useful words and phrases to take with you on holiday. x3T0 BCKs=S\.t;!THcYYX endstream WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. Minimum no. Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. /Type /ExtGState There are 6 men and 5 women in a room. stream 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. In complete bipartite graph no. Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. /Length 7 0 R WebChapter 5. Basic rules to master beginner French! /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 . 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 $w_1 + w_2 + \dots +w_m$. WebLet an = rn and substitute for all a terms to get Dividing through by rn2 to get Now we solve this polynomial using the quadratic equation Solve for r to obtain the two roots 1, 2 which is the same as A A +4 B 2 2 r= o If they are distinct, then we get o If they are the same, then we get Now apply initial conditions Graph Theory Types of Graphs 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. /Length 58 See Last Minute Notes on all subjects here. Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. @>%c0xC8a%k,s;b !AID/~ Maximum no. (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! of edges =m*n3. WebBefore tackling questions like these, let's look at the basics of counting. endobj Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. 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 $\rho_{XY}\in[-1,1]$. No. 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 Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! How many ways can you choose 3 distinct groups of 3 students from total 9 students? In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. 8"NE!OI6%pu=s{ZW"c"(E89/48q Now, it is known as the pigeonhole principle. Let q = a b and r = c d be two rational numbers written in lowest terms. How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? After filling the first and second place, (n-2) number of elements is left. /Subtype /Image U denotes the universal set. /Length 530 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 $S = \lbrace 1, 2, 3 \rbrace$. Hence, the total number of permutation is $6 \times 6 = 36$. 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 Did you make this project? 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 this looks promising :), Reply The Pigeonhole Principle 77 Chapter 6. Bnis the set of binary strings with n bits. of relations =2mn7. Let s = q + r and s = e f be written in lowest terms. A poset is called Lattice if it is both meet and join semi-lattice16. How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. Prove the following using a proof by contrapositive: Let x be a rational number. #p
Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. + \frac{ (n-1)! } /CA 1.0 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