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 $\rho_{XY} = 0$. 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} We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. 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. \(\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. 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 $^6P_{6} = 6! No. A permutation is an arrangement of some elements in which order matters. \newcommand{\va}[1]{\vtx{above}{#1}} Distributive Lattice : Every Element has zero or 1 complement .18. endobj I go out of my way to simplify subjects. Power SetsThe power set is the set all possible subset of the set S. Denoted by P(S).Example: What is the power set of {0, 1, 2}?Solution: All possible subsets{}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}.Note: Empty set and set itself is also the member of this set of subsets. How many ways are there to go from X to Z? ]8$_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. 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 i> 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 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 /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. 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 $P(n,r)\geqslant C(n,r)$. /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. /Type /Page /SMask /None>> We can also write N+= {x N : x > 0}. Boolean Lattice: It should be both complemented and distributive. \newcommand{\pow}{\mathcal P} /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. 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. \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 $n![r! Above Venn Diagram shows that A is a subset of B. >> endobj of connected components in graph with n vertices = n5. = 6$ 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. 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 << \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. Axioms of probability For each event $E$, we denote $P(E)$ as the probability of event $E$ occurring. The Pigeonhole Principle 77 Chapter 6. In how many ways we can choose 3 men and 2 women from the room? 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 From there, he can either choose 4 bus routes or 5 train routes to reach Z. Tree, 10. ("#} &. Before tackling questions like these, let's look at the basics of counting. 2195 \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} Get up and running with ChatGPT with this comprehensive cheat sheet. = 720$. \newcommand{\vr}[1]{\vtx{right}{#1}} of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . \newcommand{\R}{\mathbb R} WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <?2
AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B.To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B. \renewcommand{\v}{\vtx{above}{}} Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. ]\}$ be a partition of the sample space. No. endobj How many like both coffee and tea? % = 6$. Minimum no. WebLets dene the positive integers using the set builder notation: N+= {x : x N and x > 0}. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. 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. Learn more. >> WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3!
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