discrete math counting cheat sheet

No. Counting Then, The binomial expansion using Combinatorial symbols. 14 0 obj WebDiscrete and Combinatorial Mathematics. Discrete Mathematics \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} ]\}$ be a partition of the sample space. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe 9Wr\Gux[Eul\? How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. For complete graph the no . Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. on April 20, 2023, 5:30 PM EDT. 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. /Height 25 of irreflexive relations = 2n(n-1), 15. Counting Principles - Counting and Cardinality Pascal's identity, first derived by Blaise Pascal in 17 century, states that \newcommand{\st}{:} There are n number of ways to fill up the first place. We say that $\{A_i\}$ is a partition if we have: Remark: for any event $B$ in the sample space, we have $\displaystyle P(B)=\sum_{i=1}^nP(B|A_i)P(A_i)$. \newcommand{\amp}{&} xm=j0 gRR*9BGRGF. Definitions // Set A contains elements 1,2 and 3 A = {1,2,3} Discrete Mathematics Cheat Sheet - DocDroid Discrete Mathematics Cheat Sheet The Pigeonhole Principle 77 Chapter 6. % endobj 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 /Length 1235 Let G be a connected planar simple graph with n vertices and m edges, and no triangles. o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X %PDF-1.3 Discrete Math Cheat Sheet by Dois - Cheatography Then(a+b)modm= ((amodm) + Combinatorics 71 5.3. of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. Pascal's Identity. \newcommand{\inv}{^{-1}} /Producer ( w k h t m l t o p d f) c o m) WebDefinitions. Let s = q + r and s = e f be written in lowest terms. 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. }$, $= (n-1)! 17 0 obj In how many ways we can choose 3 men and 2 women from the room? 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. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. \newcommand{\U}{\mathcal U} 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. 1 0 obj << A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. Note that in this case it is written \mid in LaTeX, and not with the symbol |. WebProof : Assume that n is an odd integer. 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. Proof : Assume that m and n are both squares. cheat sheet 1 This is a matter of taste. There are two very important equivalences involving quantifiers. \newcommand{\Imp}{\Rightarrow} /N 100 3 0 obj 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? 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, /Type /XObject Notes on Discrete Mathematics \newcommand{\card}[1]{\left| #1 \right|} Hence, the number of subsets will be $^6C_{3} = 20$. <> 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 $c$. /ca 1.0 Question A boy lives at X and wants to go to School at Z. = 6$ ways. Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. Ten men are in a room and they are taking part in handshakes. There are $50/6 = 8$ numbers which are multiples of both 2 and 3. How many like both coffee and tea? of asymmetric relations = 3n(n-1)/211. CS160 - Fall Semester 2015. Minimum number of connected components =, 6. Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). How many ways can you distribute $10$ girl scout cookies to $7$ boy scouts? WebI COUNTING Counting things is a central problem in Discrete Mathematics. 9 years ago There must be at least two people in a class of 30 whose names start with the same alphabet. 25 0 obj << Paths and Circuits 91 3 x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] 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. Now we want to count large collections of things quickly and precisely. <> Discrete Mathematics Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. endobj It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. /Type /Page 5 0 obj 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{\gt}{>} What helped me was to take small bits of information and write them out 25 times or so. NOTE: Order of elements of a set doesnt matter. stream Heres something called a theoretical computer science cheat sheet. WebBefore tackling questions like these, let's look at the basics of counting. We make use of First and third party cookies to improve our user experience. << 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 $S = \lbrace 1, 2, 3 \rbrace$. (c) Express P(k + 1). 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 . 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 Besides, your proof of 0!=1 needs some more attention. (nr+1)! 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 /SA true Generalized Permutations and Combinations 73 5.4. ("#} &. ]$, 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 = $^np_{n}/n$. [/Pattern /DeviceRGB] (\frac{ k } { k!(n-k)! } \newcommand{\lt}{<} Simple is harder to achieve. That's a good collection you've got there, but your typesetting is aweful, I could help you with that. \newcommand{\vl}[1]{\vtx{left}{#1}} 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. See Last Minute Notes on all subjects here. FWfSE xpwy8+3o Let G be a connected planar simple graph with n vertices, where n ? WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. /Type /ObjStm \newcommand{\pow}{\mathcal P} 3 and m edges. /ImageMask true I hate discrete math because its hard for me to understand. Necessary condition for bijective function |A| = |B|5. Download the PDF version here. We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. Event Any subset $E$ of the sample space is known as an event. <> /Contents 3 0 R \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} Once we can count, we can determine the likelihood of a particular even and we can estimate how long a Representations of Graphs 88 7.3. In other words a Permutation is an ordered Combination of elements. Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. endobj Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. CME 106 - Probability Cheatsheet - Stanford University The cardinality of the set is 6 and we have to choose 3 elements from the set. The permutation will be $= 6! Size of the set S is known as Cardinality number, denoted as |S|. 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID =HA`miNDKH;8&.2_LcVfgsIVAxx$A,t([k9QR$jmOX#Q=s'0z>SUxH-5OPuVq+"a;F} Maximum no. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke No. 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. Then m 2n 4. mathematics Above Venn Diagram shows that A is a subset of B. Solution There are 6 letters word (2 E, 1 A, 1D and 2R.) In this case the sign means that a divides b, or that b a is an integer. *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! of one to one function = (n, P, m)3. 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. *3-d[\HxSi9KpOOHNn uiKa, /Contents 25 0 R /Resources 23 0 R 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. No. xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d Get up and running with ChatGPT with this comprehensive cheat sheet. 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. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. Basic rules to master beginner French! 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 Discrete mathematics cheat sheet So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. Education Cheat Sheets Assume that s is not 0. WebDiscrete Math Review n What you should know about discrete math before the midterm. 4 0 obj 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 %PDF-1.4 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. 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 combination is selection of some given elements in which order does not matter. 6 0 obj Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. of Anti Symmetric Relations = 2n*3n(n-1)/210. Here's how they described it: Equations commonly used in Discrete Math. | x |. From a night class at Fordham University, NYC, Fall, 2008. Toomey.org Tutoring Resources Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. WebChapter 5. By using this website, you agree with our Cookies Policy. 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 $n_{P_{r}}$. = 6$. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. How many anagrams are there of anagram? If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. \newcommand{\B}{\mathbf B} (b) Express P(k). of edges required = {(n-1)*(n-2)/2 } + 18. So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. We can also write N+= {x N : x > 0}. n Less theory, more problem solving, focuses on exam problems, use as study sheet! Did you make this project? 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. of spanning tree possible = nn-2. << 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 iProbability For Dummies Cheat Sheet - dummies >> endobj #p Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K \newcommand{\isom}{\cong} Every element has exactly one complement.19. A poset is called Lattice if it is both meet and join semi-lattice16. + \frac{ (n-1)! } For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. Show that if m and n are both square numbers, then m n is also a square number. Prove that if xy is irrational, then y is irrational. Corollary Let m be a positive integer and let a and b be integers. Math/CS cheat sheet. Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. Discrete Mathematics If we consider two tasks A and B which are disjoint (i.e. Probability Cheatsheet v1.1.1 Simpsons Paradox Expected /Type /Page endobj /Filter /FlateDecode stream \renewcommand{\iff}{\leftrightarrow} WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p < DISCRETE MATHEMATICS FOR COMPUTER SCIENCE /\: [(2!) 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. Cheatsheet - Summary Discrete Mathematics I Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! stream Here it means the absolute value of x, ie. / [(a_1!(a_2!) The cardinality of A B is N*M, where N is the Cardinality of A and M is the cardinality of B. UnionUnion of the sets A and B, denoted by A B, is the set of distinct element belongs to set A or set B, or both. = 180.$. Math I'll check out your sheet when I get to my computer. From his home X he has to first reach Y and then Y to Z. >> Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. /First 812 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. Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! 23 0 obj << /Length 7 0 R Complemented Lattice : Every element has complement17. $A \cap B = \emptyset$), 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. of the domain. WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? English to French cheat sheet, with useful words and phrases to take with you on holiday. WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. 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. Helps to encode it into the brain. Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. \newcommand{\Z}{\mathbb Z} 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. U denotes the universal set. WebBefore tackling questions like these, let's look at the basics of counting. Graphs 82 7.2. To guarantee that a graph with n vertices is connected, minimum no. Webdiscrete math counting cheat sheet.pdf - | Course Hero University of California, Los Angeles MATH MATH 61 discrete math counting cheat sheet.pdf - discrete math This implies that there is some integer k such that n = 2k + 1. )$. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. [Q hm*q*E9urWYN#-&\" e1cU3D).C5Q7p66[XlG|;xvvANUr_B(mVt2pzbShb5[Tv!k":,7a) A graph is euler graph if it there exists atmost 2 vertices of odd degree9. No. 8"NE!OI6%pu=s{ZW"c"(E89/48q \newcommand{\N}{\mathbb N} stream of edges to have connected graph with n vertices = n-17. }28U*~5} Kryi1#8VVN]dVOJGl\+rlN|~x lsxLw:j(b"&3X]>*~RrKa! 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. There are 6 men and 5 women in a room. { (k-1)!(n-k)! } 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 $\rho_{XY}\in[-1,1]$.