xWn7Wgv From a night class at Fordham University, NYC, Fall, 2008. The function is surjective (onto) if every element of the codomain is mapped to by at least one element. >> Proof Let there be n different elements. /SA true = 6$. { k!(n-k-1)! For complete graph the no . These are my notes created after giving the same lesson 4-5 times in one week. Above Venn Diagram shows that A is a subset of B. 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 We say that $\{A_i\}$ is a partition if we have: Remark: for any event $B$ in the sample space, we have $\displaystyle P(B)=\sum_{i=1}^nP(B|A_i)P(A_i)$. /MediaBox [0 0 612 792] Then, number of permutations of these n objects is = $n! That 3 and m edges. Then, The binomial expansion using Combinatorial symbols. endobj << of edges in a complete graph = n(n-1)/22. No. NOTE: Order of elements of a set doesnt matter. Question A boy lives at X and wants to go to School at Z. Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. | x | = { x if x 0 x if x < 0. \newcommand{\card}[1]{\left| #1 \right|} \newcommand{\vr}[1]{\vtx{right}{#1}} Now, it is known as the pigeonhole principle. Note that in this case it is written \mid in LaTeX, and not with the symbol |. It includes the enumeration or counting of objects having certain properties. \definecolor{fillinmathshade}{gray}{0.9} of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. To guarantee that a graph with n vertices is connected, minimum no. \newcommand{\Iff}{\Leftrightarrow} Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). 2 0 obj << There are 6 men and 5 women in a room. (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! Proof : Assume that m and n are both squares. Problem 2 In how many ways can the letters of the word 'READER' be arranged? %PDF-1.5 /Type /Page Necessary condition for bijective function |A| = |B|5. /Length 1781 Generalized Permutations and Combinations 73 5.4. WebCounting things is a central problem in Discrete Mathematics. @>%c0xC8a%k,s;b !AID/~ }$, $= (n-1)! In this case the sign means that a divides b, or that b a is an integer. 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists Here's how they described it: Equations commonly used in Discrete Math. So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. 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 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 edges =m*n3. /Filter /FlateDecode of relations =2mn7. 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. / [(a_1!(a_2!) Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! 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, WebChapter 5. /Type /ObjStm >> endobj 14 0 obj After filling the first and second place, (n-2) number of elements is left. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. \newcommand{\R}{\mathbb R} The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } of spanning tree possible = nn-2. on April 20, 2023, 5:30 PM EDT. \). 1 This is a matter of taste. We make use of First and third party cookies to improve our user experience. 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. &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 5 0 obj << \newcommand{\gt}{>} of functions from A to B = nm2. The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. /Height 25 ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream /Filter /FlateDecode \newcommand{\st}{:} % (b) Express P(k). /Type /ExtGState Did you make this project? Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. Basic rules to master beginner French! Get up and running with ChatGPT with this comprehensive cheat sheet. Let G be a connected planar simple graph with n vertices, where n ? We can also write N+= {x N : x > 0}. \newcommand{\amp}{&} Get up and running with ChatGPT with this comprehensive cheat sheet. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. x3T0 BCKs=S\.t;!THcYYX endstream 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 $w_1 + w_2 + \dots +w_m$. \newcommand{\imp}{\rightarrow} Combinatorics 71 5.3. 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. WebI COUNTING Counting things is a central problem in Discrete Mathematics. In other words a Permutation is an ordered Combination of elements. From there, he can either choose 4 bus routes or 5 train routes to reach Z. (1!)(1!)(2!)] +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. /SMask /None>> 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 $\rho_{XY}\in[-1,1]$. Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. There are n number of ways to fill up the first place. Affordable solution to train a team and make them project ready. }}\], \[\boxed{P(A|B)=\frac{P(B|A)P(A)}{P(B)}}\], \[\boxed{\forall i\neq j, A_i\cap A_j=\emptyset\quad\textrm{ and }\quad\bigcup_{i=1}^nA_i=S}\], \[\boxed{P(A_k|B)=\frac{P(B|A_k)P(A_k)}{\displaystyle\sum_{i=1}^nP(B|A_i)P(A_i)}}\], \[\boxed{F(x)=\sum_{x_i\leqslant x}P(X=x_i)}\quad\textrm{and}\quad\boxed{f(x_j)=P(X=x_j)}\], \[\boxed{0\leqslant f(x_j)\leqslant1}\quad\textrm{and}\quad\boxed{\sum_{j}f(x_j)=1}\], \[\boxed{F(x)=\int_{-\infty}^xf(y)dy}\quad\textrm{and}\quad\boxed{f(x)=\frac{dF}{dx}}\], \[\boxed{f(x)\geqslant0}\quad\textrm{and}\quad\boxed{\int_{-\infty}^{+\infty}f(x)dx=1}\], \[\textrm{(D)}\quad\boxed{E[X]=\sum_{i=1}^nx_if(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X]=\int_{-\infty}^{+\infty}xf(x)dx}\], \[\textrm{(D)}\quad\boxed{E[g(X)]=\sum_{i=1}^ng(x_i)f(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x)dx}\], \[\textrm{(D)}\quad\boxed{E[X^k]=\sum_{i=1}^nx_i^kf(x_i)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^k]=\int_{-\infty}^{+\infty}x^kf(x)dx}\], \[\boxed{\textrm{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2}\], \[\boxed{\sigma=\sqrt{\textrm{Var}(X)}}\], \[\textrm{(D)}\quad\boxed{\psi(\omega)=\sum_{i=1}^nf(x_i)e^{i\omega x_i}}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{\psi(\omega)=\int_{-\infty}^{+\infty}f(x)e^{i\omega x}dx}\], \[\boxed{e^{i\theta}=\cos(\theta)+i\sin(\theta)}\], \[\boxed{E[X^k]=\frac{1}{i^k}\left[\frac{\partial^k\psi}{\partial\omega^k}\right]_{\omega=0}}\], \[\boxed{f_Y(y)=f_X(x)\left|\frac{dx}{dy}\right|}\], \[\boxed{\frac{\partial}{\partial c}\left(\int_a^bg(x)dx\right)=\frac{\partial b}{\partial c}\cdot g(b)-\frac{\partial a}{\partial c}\cdot g(a)+\int_a^b\frac{\partial g}{\partial c}(x)dx}\], \[\boxed{P(|X-\mu|\geqslant k\sigma)\leqslant\frac{1}{k^2}}\], \[\textrm{(D)}\quad\boxed{f_{XY}(x_i,y_j)=P(X=x_i\textrm{ and }Y=y_j)}\], \[\textrm{(C)}\quad\boxed{f_{XY}(x,y)\Delta x\Delta y=P(x\leqslant X\leqslant x+\Delta x\textrm{ and }y\leqslant Y\leqslant y+\Delta y)}\], \[\textrm{(D)}\quad\boxed{f_X(x_i)=\sum_{j}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{f_X(x)=\int_{-\infty}^{+\infty}f_{XY}(x,y)dy}\], \[\textrm{(D)}\quad\boxed{F_{XY}(x,y)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}f_{XY}(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x',y')dx'dy'}\], \[\boxed{f_{X|Y}(x)=\frac{f_{XY}(x,y)}{f_Y(y)}}\], \[\textrm{(D)}\quad\boxed{E[X^pY^q]=\sum_{i}\sum_{j}x_i^py_j^qf(x_i,y_j)}\quad\quad\textrm{and}\quad\textrm{(C)}\quad\boxed{E[X^pY^q]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}x^py^qf(x,y)dydx}\], \[\boxed{\psi_Y(\omega)=\prod_{k=1}^n\psi_{X_k}(\omega)}\], \[\boxed{\textrm{Cov}(X,Y)\triangleq\sigma_{XY}^2=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-\mu_X\mu_Y}\], \[\boxed{\rho_{XY}=\frac{\sigma_{XY}^2}{\sigma_X\sigma_Y}}\], Distribution of a sum of independent random variables, CME 106 - Introduction to Probability and Statistics for Engineers, $\displaystyle\frac{e^{i\omega b}-e^{i\omega a}}{(b-a)i\omega}$, $\displaystyle \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, $e^{i\omega\mu-\frac{1}{2}\omega^2\sigma^2}$, $\displaystyle\frac{1}{1-\frac{i\omega}{\lambda}}$. <> 8"NE!OI6%pu=s{ZW"c"(E89/48q Note that zero is an even number, so a string. /Producer ( w k h t m l t o p d f) Let G be a connected planar simple graph with n vertices and m edges, and no triangles. = 6$ ways. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. \newcommand{\pow}{\mathcal P} 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. There must be at least two people in a class of 30 whose names start with the same alphabet. (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. A country has two political parties, the Demonstrators and the Repudiators. Representations of Graphs 88 7.3. DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream of edges required = {(n-1)*(n-2)/2 } + 18. 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

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