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discrete math counting cheat sheet

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DISCRETE MATHEMATICS FOR COMPUTER SCIENCE I dont know whether I agree with the name, but its a nice cheat sheet. )$. /Length 7 0 R 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists ]8$_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 Cardinality of power set is , where n is the number of elements in a set. Question A boy lives at X and wants to go to School at Z. Define P(n) to be the assertion that: j=1nj2=n(n+1)(2n+1)6 (a) Verify that P(3) is true. Before tackling questions like these, let's look at the basics of counting. Now, it is known as the pigeonhole principle. cheat sheet stream \newcommand{\amp}{&} In complete bipartite graph no. Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. /Resources 1 0 R \newcommand{\imp}{\rightarrow} For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. 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. The function is surjective (onto) if every element of the codomain is mapped to by at least one element. Above Venn Diagram shows that A is a subset of B. /Subtype /Image 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 <Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide /Filter /FlateDecode Learn more. 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. (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. WebProof : Assume that n is an odd integer. :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx of symmetric relations = 2n(n+1)/29. There are two very important equivalences involving quantifiers. 1 0 obj discrete math counting cheat sheet.pdf - | Course Hero Distributive Lattice : Every Element has zero or 1 complement .18. I go out of my way to simplify subjects. Discrete Mathematics - Counting Theory 1 The Rules of Sum and Product. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. 2 Permutations. A permutation is an arrangement of some elements in which order matters. 3 Combinations. 4 Pascal's Identity. 5 Pigeonhole Principle. [/Pattern /DeviceRGB] \newcommand{\N}{\mathbb N} of asymmetric relations = 3n(n-1)/211. >> endobj \newcommand{\isom}{\cong} Math }$$. >> Ten men are in a room and they are taking part in handshakes. Cheat Sheet of Mathemtical Notation and Terminology &IP")0 QlaK5 )CPq 9n TVd,L j' )3 O@ 3+$ >+:>Ov?! | x |. WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. Representations of Graphs 88 7.3. Mathematics | Combinatorics Basics Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. 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. 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. \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} << Discrete Math Cheat Sheet by Dois - Cheatography %PDF-1.4 Then(a+b)modm= ((amodm) + $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. By noting $f$ and $F$ the PDF and CDF respectively, we have the following relations: Continuous case Here, $X$ takes continuous values, such as the temperature in the room. 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 I have a class in it right now actually! of edges =m*n3. WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. >> For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? Paths and Circuits 91 3 28 0 obj << No. xWn7Wgv o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X 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. Discrete Mathematics WebBefore tackling questions like these, let's look at the basics of counting. There must be at least two people in a class of 30 whose names start with the same alphabet. In this case it is written with just the | symbol. 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)$. 9 years ago >> endobj Harold's Cheat Sheets "If you can't explain it simply, you don't understand it well enough." Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. This ordered or stable list of counting words must be at least as long as the number of items to be counted. /Type /ExtGState /First 812 Discrete Mathematics xY8_1ow>;|D@`a%e9l96=u=uQ /Length 1781 Every element has exactly one complement.19. Heres something called a theoretical computer science cheat sheet. Web445 Cheatsheet. No. &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 A combination is selection of some given elements in which order does not matter. Hence, the total number of permutation is $6 \times 6 = 36$. 17 0 obj 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. *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! Probability Cheatsheet v1.1.1 Simpsons Paradox Expected \newcommand{\C}{\mathbb C} 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. 3 0 obj << +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. 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)$. { r!(n-r)! In this case the sign means that a divides b, or that b a is an integer. In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. Bipartite Graph : There is no edges between any two vertices of same partition . >> endobj Complemented Lattice : Every element has complement17. Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. Then, number of permutations of these n objects is = $n! In other words a Permutation is an ordered Combination of elements. \newcommand{\st}{:} \newcommand{\vr}[1]{\vtx{right}{#1}} Size of a SetSize of a set can be finite or infinite. /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 . /MediaBox [0 0 612 792] Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. WebDiscrete Math Review n What you should know about discrete math before the midterm. ~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. on April 20, 2023, 5:30 PM EDT. 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$. Web2362 Education Cheat Sheets. Education Cheat Sheets \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} WebBefore tackling questions like these, let's look at the basics of counting. By using this website, you agree with our Cookies Policy. If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. WebDefinitions. 2195 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 . Notes on Discrete Mathematics element of the domain. /SM 0.02 A relation is an equivalence if, 1. stream 2 0 obj << /Width 156 6 0 obj 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID =HA`miNDKH;8&.2_LcVfgsIVAxx$A,t([k9QR$jmOX#Q=s'0z>SUxH-5OPuVq+"a;F} Last Minute Notes Discrete Mathematics - GeeksforGeeks Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. In how many ways we can choose 3 men and 2 women from the room? endobj English to French cheat sheet, with useful words and phrases to take with you on holiday. xKs6. Binomial Coecients 75 5.5. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! 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. CPS102 DISCRETE MATHEMATICS Practice Final Exam - Duke /SA true | x | = { x if x 0 x if x < 0. ]$, 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$. Generalized Permutations and Combinations 73 5.4. Probability 78 6.1. I'll check out your sheet when I get to my computer. CME 106 - Probability Cheatsheet - Stanford University WebThe first principle of counting involves the student using a list of words to count in a repeatable order. From a night class at Fordham University, NYC, Fall, 2008. %PDF-1.3 Here, the ordering does not matter. (1!)(1!)(2!)] The permutation will be = 123, 132, 213, 231, 312, 321, The number of permutations of n different things taken r at a time is denoted by $n_{P_{r}}$. Below is a quick refresher on some math tools and problem-solving techniques from 240 (or other prereqs) that well assume knowledge of for the PSets. <> /Filter /FlateDecode /ImageMask true 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$. WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . 1 0 obj << Cartesian ProductsLet A and B be two sets. Now we want to count large collections of things quickly and precisely. >> endobj 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. of one to one function = (n, P, m)3. Assume that s is not 0. Simple is harder to achieve. 5 0 obj << Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d /Font << /F17 6 0 R /F18 9 0 R /F15 12 0 R /F7 15 0 R /F8 18 0 R /F37 21 0 R >> So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. \newcommand{\U}{\mathcal U} \newcommand{\vb}[1]{\vtx{below}{#1}} How many integers from 1 to 50 are multiples of 2 or 3 but not both? Hence, the number of subsets will be $^6C_{3} = 20$. 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 iCounting - Discrete Mathematics He may go X to Y by either 3 bus routes or 2 train routes. 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 Course Hero is not sponsored or endorsed by any college or university. \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. (c) Express P(k + 1). on April 20, 2023, 5:30 PM EDT. \renewcommand{\v}{\vtx{above}{}} Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). /ProcSet [ /PDF ] Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. What helped me was to take small bits of information and write them out 25 times or so. Here it means the absolute value of x, ie. 25 0 obj << 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. 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. (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! This implies that there is some integer k such that n = 2k + 1. Prove the following using a proof by contrapositive: Let x be a rational number. That mathematics BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. 592 /Parent 22 0 R of functions from A to B = nm2. /AIS false A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. For solving these problems, mathematical theory of counting are used. \renewcommand{\iff}{\leftrightarrow} Discrete Math Cram Sheet - Ateneo de Manila University Tree, 10. WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. '1g[bXlF) q^|W*BmHYGd tK5A+(R%9;P@2[P9?j28C=r[%\%U08$@`TaqlfEYCfj8Zx!`,O%L v+ ]F$Dx U. 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. I strongly believe that simple is better than complex. Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). 5 0 obj \[\boxed{P\left(\bigcup_{i=1}^nE_i\right)=\sum_{i=1}^nP(E_i)}\], \[\boxed{C(n, r)=\frac{P(n, r)}{r!}=\frac{n!}{r!(n-r)! 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. Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? Hence, there are 10 students who like both tea and coffee. /Length 530 Extended form of Bayes' rule Let $\{A_i, i\in[\![1,n]\! stream *3-d[\HxSi9KpOOHNn uiKa, \newcommand{\R}{\mathbb 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? /Height 25 Discrete Mathematics Cheat Sheet - DocDroid Once we can count, we can determine the likelihood of a particular even and we can estimate how long a Share it with us! endobj Proof : Assume that m and n are both squares. Math/CS cheat sheet. /Filter /FlateDecode /MediaBox [0 0 612 792] <> Get up and running with ChatGPT with this comprehensive cheat sheet. stream /Resources 23 0 R A poset is called Lattice if it is both meet and join semi-lattice16. Discrete Mathematics of the domain. of Anti Symmetric Relations = 2n*3n(n-1)/210. 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. Discrete Mathematics Cheat Sheet \newcommand{\Q}{\mathbb Q} 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. endobj = 6$. Counting 69 5.1. Webdiscrete math counting cheat sheet.pdf - | Course Hero University of California, Los Angeles MATH MATH 61 discrete math counting cheat sheet.pdf - discrete math DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream There are $50/3 = 16$ numbers which are multiples of 3. /Length 58 I hate discrete math because its hard for me to understand. xm=j0 gRR*9BGRGF. The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } 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. 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 Download the PDF version here. How many anagrams are there of anagram? %PDF-1.2 Show that if m and n are both square numbers, then m n is also a square number. Solution There are 6 letters word (2 E, 1 A, 1D and 2R.) Probability For Dummies Cheat Sheet - dummies Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). 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. U denotes the universal set. Hi matt392, nice work! How many ways can you choose 3 distinct groups of 3 students from total 9 students? Types of propositions based on Truth values1.Tautology A proposition which is always true, is called a tautology.2.Contradiction A proposition which is always false, is called a contradiction.3.Contingency A proposition that is neither a tautology nor a contradiction is called a contingency. 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. 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. Then m 2n 4. 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. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke >> \newcommand{\B}{\mathbf B} 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! endobj stream \newcommand{\gt}{>} Reference Sheet for Discrete Maths - GitHub Pages These are my notes created after giving the same lesson 4-5 times in one week. { 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} We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. Hence, there are (n-1) ways to fill up the second place. One of the first things you learn in mathematics is how to count. 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 ("#} &. Note that zero is an even number, so a string. Basic Principles 69 5.2. Owens Funeral Home Cartersville, Ga, Steroid Shot In Buttocks For Sinus Infection, Sports Cardiologist Miami, Pastor Insults Church Members, Has The Trumpet Been Blown, Articles D
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