\YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. Counting - Discrete Mathematics endobj <> \newcommand{\vl}[1]{\vtx{left}{#1}} [/Pattern /DeviceRGB] Graph Theory 82 7.1. ~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. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE Get up and running with ChatGPT with this comprehensive cheat sheet. WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <Cheat Sheet Once we can count, we can determine the likelihood of a particular even and we can estimate how long a The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } Mathematics | Combinatorics Basics Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. Discrete Math Review stream You can use all your notes, calcu-lator, and any books you >> \definecolor{fillinmathshade}{gray}{0.9} 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 WebI COUNTING Counting things is a central problem in Discrete Mathematics. endobj /Length 530 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. Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. Hence, the total number of permutation is $6 \times 6 = 36$. *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! Minimum no. Discrete Mathematics 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! No. endobj Maximum no. No. No. How many like both coffee and tea? Define the set Ento be the set of binary strings with n bits that have an even number of 1's. Cardinality of power set is , where n is the number of elements in a set. /Length 1781 For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? /Length 58 There are $50/6 = 8$ numbers which are multiples of both 2 and 3. endobj Counting Principles - Counting and Cardinality 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. endobj 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. No. WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. stream Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. /CreationDate (D:20151115165753Z) xWn7Wgv 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 $\rho_{XY}\in[-1,1]$. endobj No. 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 iSUxH-5OPuVq+"a;F} /Type /XObject Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. endobj \). /SMask /None>> { (k-1)!(n-k)! } \newcommand{\Q}{\mathbb Q} on April 20, 2023, 5:30 PM EDT. (\frac{ k } { k!(n-k)! } What helped me was to take small bits of information and write them out 25 times or so. Counting 69 5.1. Let q = a b and r = c d be two rational numbers written in lowest terms. Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. Probability For Dummies Cheat Sheet - dummies Then, The binomial expansion using Combinatorial symbols. Discrete mathematics cheat sheet For solving these problems, mathematical theory of counting are used. Discrete Mathematics Cheat Sheet - DocDroid We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. Basic Principles 69 5.2. Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. stream This implies that there is some integer k such that n = 2k + 1. 1 This is a matter of taste. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe
9Wr\Gux[Eul\? }$, $= (n-1)! No. 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists 3 0 obj << From a night class at Fordham University, NYC, Fall, 2008. )$. How many ways can you choose 3 distinct groups of 3 students from total 9 students? Vertical bar sign in Discrete mathematics Generalized Permutations and Combinations 73 5.4. 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. There are two very important equivalences involving quantifiers. It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. Number of permutations of n distinct elements taking n elements at a time = $n_{P_n} = n!$, The number of permutations of n dissimilar elements taking r elements at a time, when x particular things always occupy definite places = $n-x_{p_{r-x}}$, The number of permutations of n dissimilar elements when r specified things always come together is $r! \renewcommand{\bar}{\overline} Cartesian ProductsLet A and B be two sets. Minimum number of connected components =, 6. /Type /ExtGState &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 / [(a_1!(a_2!) 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 That is, an event is a set consisting of possible outcomes of the experiment. Get up and running with ChatGPT with this comprehensive cheat sheet. 5 0 obj In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. Hence, the number of subsets will be $^6C_{3} = 20$. A combination is selection of some given elements in which order does not matter. Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. Equivalesistheonlyequivalencerelationthatisassociative ((p q) r) (p (q Show that if m and n are both square numbers, then m n is also a square number. Math 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 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. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. /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 . 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. /Creator () \newcommand{\Z}{\mathbb Z} \newcommand{\B}{\mathbf B} Ten men are in a room and they are taking part in handshakes. \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} Prove or disprove the following two statements. That's a good collection you've got there, but your typesetting is aweful, I could help you with that. \newcommand{\U}{\mathcal U} Share it with us! For $k, \sigma>0$, we have the following inequality: Discrete distributions Here are the main discrete distributions to have in mind: Continuous distributions Here are the main continuous distributions to have in mind: Joint probability density function The joint probability density function of two random variables $X$ and $Y$, that we note $f_{XY}$, is defined as follows: Marginal density We define the marginal density for the variable $X$ as follows: Cumulative distribution We define cumulative distrubution $F_{XY}$ as follows: Conditional density The conditional density of $X$ with respect to $Y$, often noted $f_{X|Y}$, is defined as follows: Independence Two random variables $X$ and $Y$ are said to be independent if we have: Moments of joint distributions We define the moments of joint distributions of random variables $X$ and $Y$ as follows: Distribution of a sum of independent random variables Let $Y=X_1++X_n$ with $X_1, , X_n$ independent. One of the first things you learn in mathematics is how to count. There are n number of ways to fill up the first place. No. Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$. Probability 78 Chapter 7. /SA true Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). WebChapter 5. Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. + \frac{ n-k } { k!(n-k)! } 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
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