Get up and running with ChatGPT with this comprehensive cheat sheet. (\frac{ k } { k!(n-k)! } Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. Math WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. \newcommand{\Iff}{\Leftrightarrow} DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. Counting No. /Resources 23 0 R Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). Did you make this project? + \frac{ n-k } { k!(n-k)! } >> 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 Partition Let $\{A_i, i\in[\![1,n]\! Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). Let s = q + r and s = e f be written in lowest terms. endobj How many ways can you choose 3 distinct groups of 3 students from total 9 students? )$. Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE \(\renewcommand{\d}{\displaystyle} /CreationDate (D:20151115165753Z) gQVmDYm*%
QKP^n,D%7DBZW=pvh#(sG See Last Minute Notes on all subjects here. To guarantee that a graph with n vertices is connected, minimum no. stream Heres something called a theoretical computer science cheat sheet. xm=j0 gRR*9BGRGF. Cheatsheet - Summary Discrete Mathematics I of one to one function = (n, P, m)3. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! No. A permutation is an arrangement of some elements in which order matters. \newcommand{\va}[1]{\vtx{above}{#1}} Distributive Lattice : Every Element has zero or 1 complement .18. endobj I go out of my way to simplify subjects. Power SetsThe power set is the set all possible subset of the set S. Denoted by P(S).Example: What is the power set of {0, 1, 2}?Solution: All possible subsets{}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}.Note: Empty set and set itself is also the member of this set of subsets. Discrete Math Cheat Sheet by Dois - Cheatography How many ways are there to go from X to Z? ]8$_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 This implies that there is some integer k such that n = 2k + 1. He may go X to Y by either 3 bus routes or 2 train routes. of asymmetric relations = 3n(n-1)/211. Cheat Sheet 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, Now we want to count large collections of things quickly and precisely. Harold's Cheat Sheets "If you can't explain it simply, you don't understand it well enough." / [(a_1!(a_2!) Hi matt392, nice work! stream WebThe first principle of counting involves the student using a list of words to count in a repeatable order. on April 20, 2023, 5:30 PM EDT. By using this website, you agree with our Cookies Policy. Pascal's identity, first derived by Blaise Pascal in 17 century, states that 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 /Filter /FlateDecode Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. Note that in this case it is written \mid in LaTeX, and not with the symbol |. Education Cheat Sheets WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . 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 $\rho_{XY}\in[-1,1]$. of Anti Symmetric Relations = 2n*3n(n-1)/210. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). Hence, the number of subsets will be $^6C_{3} = 20$. endobj \newcommand{\Q}{\mathbb Q} Prove the following using a proof by contrapositive: Let x be a rational number. WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. % Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } I'll check out your sheet when I get to my computer. Representations of Graphs 88 7.3. Expected value The expected value of a random variable, also known as the mean value or the first moment, is often noted $E[X]$ or $\mu$ and is the value that we would obtain by averaging the results of the experiment infinitely many times. %PDF-1.4 Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional Paths and Circuits 91 3 Complemented Lattice : Every element has complement17. ];_. %PDF-1.3 of reflexive relations =2n(n-1)8. \newcommand{\vb}[1]{\vtx{below}{#1}} 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. 2 0 obj << /ProcSet [ /PDF /Text ] /Contents 3 0 R 6 0 obj of spanning tree possible = nn-2. Here's how they described it: Equations commonly used in Discrete Math. A country has two political parties, the Demonstrators and the Repudiators. ]$, 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 = $^np_{n}/n$. 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. Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. /\: [(2!) Counting Principles - Counting and Cardinality /Width 156 WebI COUNTING Counting things is a central problem in Discrete Mathematics. /Type /ExtGState << \renewcommand{\iff}{\leftrightarrow} of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. Show that if m and n are both square numbers, then m n is also a square number. 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. of bijection function =n!6. >> of edges required = {(n-1)*(n-2)/2 } + 18. NOTE: Order of elements of a set doesnt matter. on Introduction. The function is injective (one-to-one) if every element of the codomain is mapped to by at most one. 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. This ordered or stable list of counting words must be at least as long as the number of items to be counted. Hence, the total number of permutation is $6 \times 6 = 36$. Ten men are in a room and they are taking part in handshakes. of edges to have connected graph with n vertices = n-17. \[\boxed{P\left(\bigcup_{i=1}^nE_i\right)=\sum_{i=1}^nP(E_i)}\], \[\boxed{C(n, r)=\frac{P(n, r)}{r!}=\frac{n!}{r!(n-r)! /First 812 xWn7Wgv 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. Extended form of Bayes' rule Let $\{A_i, i\in[\![1,n]\! WebDefinitions. stream Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. There must be at least two people in a class of 30 whose names start with the same alphabet. acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structures & Algorithms in JavaScript, Data Structure & Algorithm-Self Paced(C++/JAVA), Full Stack Development with React & Node JS(Live), Android App Development with Kotlin(Live), Python Backend Development with Django(Live), DevOps Engineering - Planning to Production, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Mathematics | Introduction to Propositional Logic | Set 1, Discrete Mathematics Applications of Propositional Logic, Difference between Propositional Logic and Predicate Logic, Mathematics | Predicates and Quantifiers | Set 1, Mathematics | Some theorems on Nested Quantifiers, Mathematics | Set Operations (Set theory), Mathematics | Sequence, Series and Summations, Mathematics | Representations of Matrices and Graphs in Relations, Mathematics | Introduction and types of Relations, Mathematics | Closure of Relations and Equivalence Relations, Permutation and Combination Aptitude Questions and Answers, Discrete Maths | Generating Functions-Introduction and Prerequisites, Inclusion-Exclusion and its various Applications, Project Evaluation and Review Technique (PERT), Mathematics | Partial Orders and Lattices, Mathematics | Probability Distributions Set 1 (Uniform Distribution), Mathematics | Probability Distributions Set 2 (Exponential Distribution), Mathematics | Probability Distributions Set 3 (Normal Distribution), Mathematics | Probability Distributions Set 5 (Poisson Distribution), Mathematics | Graph Theory Basics Set 1, Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph, Mathematics | Independent Sets, Covering and Matching, How to find Shortest Paths from Source to all Vertices using Dijkstras Algorithm, Introduction to Tree Data Structure and Algorithm Tutorials, Prims Algorithm for Minimum Spanning Tree (MST), Kruskals Minimum Spanning Tree (MST) Algorithm, Tree Traversals (Inorder, Preorder and Postorder), Travelling Salesman Problem using Dynamic Programming, Check whether a given graph is Bipartite or not, Eulerian path and circuit for undirected graph, Fleurys Algorithm for printing Eulerian Path or Circuit, Chinese Postman or Route Inspection | Set 1 (introduction), Graph Coloring | Set 1 (Introduction and Applications), Check if a graph is Strongly, Unilaterally or Weakly connected, Handshaking Lemma and Interesting Tree Properties, Mathematics | Rings, Integral domains and Fields, Topic wise multiple choice questions in computer science, A graph is planar if and only if it does not contain a subdivision of K. Let G be a connected planar graph, and let n, m and f denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of G. Then n m + f = 2. 28 0 obj << stream discrete math counting cheat sheet.pdf - | Course Hero Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. By using our site, you CME 106 - Probability Cheatsheet - Stanford University /Filter /FlateDecode '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} Graph Theory 82 7.1. 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 $c62MC*u+Z In other words a Permutation is an ordered Combination of elements. WebBefore tackling questions like these, let's look at the basics of counting. 592 Proof Let there be n different elements. Helps to encode it into the brain. Probability 78 Chapter 7. xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF
d Cardinality of power set is , where n is the number of elements in a set. /Length 58 \dots (a_r!)]$. Hence, a+c b+d(modm)andac bd(modm). | x |. The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! /ImageMask true /Filter /FlateDecode >> Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. ChatGPT cheat sheet: Complete guide for 2023 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 $P(n,r)\geqslant C(n,r)$. Last Minute Notes Discrete Mathematics - GeeksforGeeks /Parent 22 0 R There are 6 men and 5 women in a room. 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 $n_{P_{r}}$. 9 years ago >> endobj /AIS false 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. Minimum number of connected components =, 6. \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} 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. 1 0 obj << \newcommand{\Imp}{\Rightarrow} in the word 'READER'. In this case it is written with just the | symbol. \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. Then m 3n 6. @>%c0xC8a%k,s;b !AID/~ BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. /Subtype /Image Define the set Ento be the set of binary strings with n bits that have an even number of 1's. Discrete Mathematics /Type /Page /SMask /None>> We can also write N+= {x N : x > 0}. Boolean Lattice: It should be both complemented and distributive. Discrete Mathematics Cheat Sheet \newcommand{\pow}{\mathcal P} Reference Sheet for Discrete Maths - GitHub Pages /Contents 25 0 R WebDiscrete and Combinatorial Mathematics. Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). No. Event Any subset $E$ of the sample space is known as an event. The permutation will be $= 6! Graph Theory; Notes on Counting; Notes on Distributions and Stirling numbers of the second kind; Notes on Cardinality of Sets; Notes on the Pigeonhole Principle; Notes on Combinatorial Arguments; Notes on Recurrence Relations; Notes on Inclusion-Exclusion; Notes on Generating Functions 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. A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. A graph is euler graph if it there exists atmost 2 vertices of odd degree9. stream No. of the domain. Discrete Mathematics 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. /Type /ObjStm Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. \newcommand{\vl}[1]{\vtx{left}{#1}} xKs6. Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide \newcommand{\N}{\mathbb N} If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. <> WebBefore tackling questions like these, let's look at the basics of counting. element of the domain. 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. /ca 1.0 { k!(n-k-1)! What helped me was to take small bits of information and write them out 25 times or so. Necessary condition for bijective function |A| = |B|5. /Length 7 0 R +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. 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} That's a good collection you've got there, but your typesetting is aweful, I could help you with that. Affordable solution to train a team and make them project ready. /Filter /FlateDecode endobj For complete graph the no . /Type /XObject ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream Every element has exactly one complement.19. $A \cap B = \emptyset$), 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. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. c o m) \newcommand{\inv}{^{-1}} \newcommand{\B}{\mathbf B} If we consider two tasks A and B which are disjoint (i.e. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. SA+9)UI)bwKJGJ-4D
tFX9LQ Web445 Cheatsheet. 6 0 obj It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. Size of the set S is known as Cardinality number, denoted as |S|. Simple is harder to achieve. >> endobj There are $50/3 = 16$ numbers which are multiples of 3. | x | = { x if x 0 x if x < 0. >> Assume that s is not 0. \newcommand{\amp}{&} :oCH7ZG_
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?K?*]ZrLbu7,J^(80~*@dL"rjx (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! Above Venn Diagram shows that A is a subset of B. >> endobj of connected components in graph with n vertices = n5. CPS102 DISCRETE MATHEMATICS Practice Final Exam - Duke = 6$ ways. 1 This is a matter of taste. 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 /Length 1235 WebDiscrete Math Review n What you should know about discrete math before the midterm. There are two very important equivalences involving quantifiers. WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. Besides, your proof of 0!=1 needs some more attention. We say that $\{A_i\}$ is a partition if we have: Remark: for any event $B$ in the sample space, we have $\displaystyle P(B)=\sum_{i=1}^nP(B|A_i)P(A_i)$. 3 and m edges. of edges =m*n3. Toomey.org Tutoring Resources I hate discrete math because its hard for me to understand. Problem 2 In how many ways can the letters of the word 'READER' be arranged? \newcommand{\card}[1]{\left| #1 \right|} Web2362 Education Cheat Sheets. Prove that if xy is irrational, then y is irrational. How many integers from 1 to 50 are multiples of 2 or 3 but not both? /MediaBox [0 0 612 792] Pascal's Identity. How many anagrams are there of anagram? xY8_1ow>;|D@`a%e9l96=u=uQ { (k-1)!(n-k)! } No. Probability 78 6.1. Let q = a b and r = c d be two rational numbers written in lowest terms. 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 $(a_1 + a_2 + a_r) = n$. '1g[bXlF) q^|W*BmHYGd tK5A+(R%9;P@2[P9?j28C=r[%\%U08$@`TaqlfEYCfj8Zx!`,O%L v+ ]F$Dx U. 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. Bnis the set of binary strings with n bits. (b) Express P(k). 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 Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. Counting 69 5.1. Definitions // Set A contains elements 1,2 and 3 A = {1,2,3} Define P(n) to be the assertion that: j=1nj2=n(n+1)(2n+1)6 (a) Verify that P(3) is true. We make use of First and third party cookies to improve our user experience. One of the first things you learn in mathematics is how to count. Prove or disprove the following two statements. stream For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. /Filter /FlateDecode \newcommand{\U}{\mathcal U} endobj The function is surjective (onto) if every element of the codomain is mapped to by at least one element. /Resources 1 0 R this looks promising :), Reply on April 20, 2023, 5:30 PM EDT. + \frac{ (n-1)! } /Parent 22 0 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? U denotes the universal set. 25 0 obj << Discrete Structures Lecture Notes - Stanford University \newcommand{\C}{\mathbb C} Webdiscrete math counting cheat sheet.pdf - | Course Hero University of California, Los Angeles MATH MATH 61 discrete math counting cheat sheet.pdf - discrete math /Height 25 Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. /N 100 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 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.
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