WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed.
Education Cheat Sheets Simple is harder to achieve. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! Note that in this case it is written \mid in LaTeX, and not with the symbol |. For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. This implies that there is some integer k such that n = 2k + 1. >> stream + \frac{ n-k } { k!(n-k)! } /SM 0.02 Then(a+b)modm= ((amodm) + 6 0 obj 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} WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. 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. xY8_1ow>;|D@`a%e9l96=u=uQ \[\boxed{P\left(\bigcup_{i=1}^nE_i\right)=\sum_{i=1}^nP(E_i)}\], \[\boxed{C(n, r)=\frac{P(n, r)}{r!}=\frac{n!}{r!(n-r)! of edges =m*n3. \dots (a_r!)]$. <> ~C'ZOdA3,3FHaD%B,e@,*/x}9Scv\`{]SL*|)B(u9V|My\4 Xm$qg3~Fq&M?D'Clk +&$.U;n8FHCfQd!gzMv94NU'M`cU6{@zxG,,?F,}I+52XbQN0.''f>:Vn(g."]^{\p5,`"zI%nO. Graphs 82 7.2. WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . A relation is an equivalence if, 1. English to French cheat sheet, with useful words and phrases to take with you on holiday. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe
9Wr\Gux[Eul\? 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 NOTE: Order of elements of a set doesnt matter. WebCounting things is a central problem in Discrete Mathematics. CS160 - Fall Semester 2015. Representations of Graphs 88 7.3. The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! In how many ways we can choose 3 men and 2 women from the room? x3T0 BCKs=S\.t;!THcYYX endstream
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WebI COUNTING Counting things is a central problem in Discrete Mathematics.
Discrete Math Review We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. /Length 7 0 R We say that $\{A_i\}$ is a partition if we have: Remark: for any event $B$ in the sample space, we have $\displaystyle P(B)=\sum_{i=1}^nP(B|A_i)P(A_i)$. stream So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. 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. Basic Principles 69 5.2. of connected components in graph with n vertices = n5. From his home X he has to first reach Y and then Y to Z. 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. BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?.
Thank you - hope it helps. Hence, there are (n-1) ways to fill up the second place. \newcommand{\Imp}{\Rightarrow} SA+9)UI)bwKJGJ-4D
tFX9LQ 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 . No. U denotes the universal set. >> endobj stream \renewcommand{\bar}{\overline} Bnis the set of binary strings with n bits. 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. /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 . 4 0 obj WebChapter 5. /Filter /FlateDecode << Here, the ordering does not matter. By using this website, you agree with our Cookies Policy. 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. }28U*~5} Kryi1#8VVN]dVOJGl\+rlN|~x lsxLw:j(b"&3X]>*~RrKa! of edges in a complete graph = n(n-1)/22. endobj Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. >> A poset is called Lattice if it is both meet and join semi-lattice16. No. \newcommand{\lt}{<} 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
o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X WebTrig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <
xWn7Wgv of irreflexive relations = 2n(n-1), 15. 9 years ago \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. WebProof : Assume that n is an odd integer. How many ways can you choose 3 distinct groups of 3 students from total 9 students? To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B.To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B. /ImageMask true Harold's Cheat Sheets "If you can't explain it simply, you don't understand it well enough." WebThe first principle of counting involves the student using a list of words to count in a repeatable order. Toomey.org Tutoring Resources Get up and running with ChatGPT with this comprehensive cheat sheet. Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, /N 100 Generalized Permutations and Combinations 73 5.4. of symmetric relations = 2n(n+1)/29. Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. /MediaBox [0 0 612 792] Cartesian product of A and B is denoted by A B, is the set of all ordered pairs (a, b), where a belong to A and b belong to B. \newcommand{\amp}{&} /Resources 23 0 R Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. Equal setsTwo sets are said to be equal if both have same elements. /Length 1235 \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} 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. 1 0 obj \newcommand{\inv}{^{-1}} of Anti Symmetric Relations = 2n*3n(n-1)/210. /SMask /None>> Let s = q + r and s = e f be written in lowest terms. stream 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 gQVmDYm*%
QKP^n,D%7DBZW=pvh#(sG >> 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. Discrete Math Cheat Sheet by Dois #education #mathematics #math \newcommand{\imp}{\rightarrow} In complete bipartite graph no. >> Every element has exactly one complement.19. 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. Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). It includes the enumeration or counting of objects having certain properties. 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 $P(n,r)\geqslant C(n,r)$. From a night class at Fordham University, NYC, Fall, 2008. 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. He may go X to Y by either 3 bus routes or 2 train routes. 17 0 obj FWfSE
xpwy8+3o 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 $w_1 + w_2 + \dots +w_m$. Complemented Lattice : Every element has complement17. A country has two political parties, the Demonstrators and the Repudiators. ChatGPT cheat sheet: Complete guide for 2023 = 6$. Download the PDF version here. Discrete mathematics cheat sheet /CreationDate (D:20151115165753Z) 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 WebDiscrete Math Review n What you should know about discrete math before the midterm. endobj 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. The number of all combinations of n things, taken r at a time is , $$^nC_{ { r } } = \frac { n! } No. How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. Probability Cheatsheet v2.0 Thinking Conditionally Law of >> endobj If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. xKs6. I have a class in it right now actually! /Type /Page A combination is selection of some given elements in which order does not matter. 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 $S = \lbrace 1, 2, 3 \rbrace$. Prove that if xy is irrational, then y is irrational. 592 <> \newcommand{\R}{\mathbb R} DISCRETE MATHEMATICS FOR COMPUTER SCIENCE 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! Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. Get up and running with ChatGPT with this comprehensive cheat sheet. >> endobj 6 0 obj Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. | x |. 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. \newcommand{\va}[1]{\vtx{above}{#1}} Let G be a connected planar simple graph with n vertices, where n ? / [(a_1!(a_2!) 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. of bijection function =n!6. This ordered or stable list of counting words must be at least as long as the number of items to be counted. Share it with us! Pascal's identity, first derived by Blaise Pascal in 17 century, states that WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. 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. Heres something called a theoretical computer science cheat sheet. 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists /\: [(2!) Helps to encode it into the brain. of spanning tree possible = nn-2. 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. Prove the following using a proof by contrapositive: Let x be a rational number. Thus, n2 is odd. \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} { (k-1)!(n-k)! } How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? \newcommand{\Q}{\mathbb Q} discrete math counting cheat sheet.pdf - | Course Hero endobj of relations =2mn7. \(\renewcommand{\d}{\displaystyle} From there, he can either choose 4 bus routes or 5 train routes to reach Z. >> Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? 3 0 obj << Besides, your proof of 0!=1 needs some more attention. stream Proof Let there be n different elements. Assume that s is not 0. Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } Cardinality of power set is , where n is the number of elements in a set. Mathematics | Combinatorics Basics stream xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF
d Bayes' rule For events $A$ and $B$ such that $P(B)>0$, we have: Remark: we have $P(A\cap B)=P(A)P(B|A)=P(A|B)P(B)$. Then, number of permutations of these n objects is = $n! Before tackling questions like these, let's look at the basics of counting. /Type /Page of reflexive relations =2n(n-1)8. Discrete Math Cheat Sheet by Dois - Cheatography = 6$ ways. No. After filling the first place (n-1) number of elements is left. ]\}$ be a partition of the sample space. 1 0 obj << 14 0 obj So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. Math /ca 1.0 %PDF-1.2 2195 Cheat Sheet of Mathemtical Notation and Terminology If we consider two tasks A and B which are disjoint (i.e. Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. endobj \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} Now we want to count large collections of things quickly and precisely. }$$. Distributive Lattice : Every Element has zero or 1 complement .18. Then m 3n 6. \newcommand{\vr}[1]{\vtx{right}{#1}} WebBefore tackling questions like these, let's look at the basics of counting. In this case it is written with just the | symbol. /Filter /FlateDecode Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). How many anagrams are there of anagram? 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 Hence, a+c b+d(modm)andac bd(modm). We have: Independence Two events $A$ and $B$ are independent if and only if we have: Random variable A random variable, often noted $X$, is a function that maps every element in a sample space to a real line. Hence, the total number of permutation is $6 \times 6 = 36$. +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. 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 $r! << I strongly believe that simple is better than complex. Discrete Math Cram Sheet - Ateneo de Manila University Combinatorics 71 5.3. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke Math/CS cheat sheet. of functions from A to B = nm2. /ProcSet [ /PDF ] See Last Minute Notes on all subjects here. of the domain. <> The cardinality of A B is N*M, where N is the Cardinality of A and M is the cardinality of B. UnionUnion of the sets A and B, denoted by A B, is the set of distinct element belongs to set A or set B, or both. Last Minute Notes Discrete Mathematics - GeeksforGeeks xm=j0 gRR*9BGRGF. Discrete Mathematics DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream { r!(n-r)! /First 812 Discrete Math 1: Set Theory. Cheat Sheet | by Alex Roan - Medium 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 $\rho_{XY}\in[-1,1]$. ("#} &. { k!(n-k-1)! 28 0 obj << Reference Sheet for Discrete Maths - GitHub Pages There are two very important equivalences involving quantifiers. Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. )$. WebDiscrete and Combinatorial Mathematics. = 180.$. 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. The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. Axiom 1 Every probability is between 0 and 1 included, i.e: Axiom 2 The probability that at least one of the elementary events in the entire sample space will occur is 1, i.e: Axiom 3 For any sequence of mutually exclusive events $E_1, , E_n$, we have: Permutation A permutation is an arrangement of $r$ objects from a pool of $n$ objects, in a given order. How many like both coffee and tea? Discrete Mathematics Cheat Sheet Hence, the number of subsets will be $^6C_{3} = 20$. stream on April 20, 2023, 5:30 PM EDT. a b. Then m 2n 4. Axioms of probability For each event $E$, we denote $P(E)$ as the probability of event $E$ occurring. Basic rules to master beginner French! Let G be a connected planar simple graph with n vertices and m edges, and no triangles. Show that if m and n are both square numbers, then m n is also a square number. Graph Theory 82 7.1. These are my notes created after giving the same lesson 4-5 times in one week. We make use of First and third party cookies to improve our user experience. Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Vertical bar sign in Discrete mathematics A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. Tree, 10. ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream /Creator () There are 6 men and 5 women in a room. Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. There are $50/3 = 16$ numbers which are multiples of 3. \newcommand{\vb}[1]{\vtx{below}{#1}} element of the domain. What helped me was to take small bits of information and write them out 25 times or so. endobj We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. 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, That /Parent 22 0 R \renewcommand{\iff}{\leftrightarrow} Minimum number of connected components =, 6. Discrete Mathematics Cheat Sheet - DocDroid Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! After filling the first and second place, (n-2) number of elements is left. Probability For Dummies Cheat Sheet - dummies Counting - Discrete Mathematics Probability 78 6.1. #p
Na~ Z&+K@"SLr4!rb1J"\]d``xMl-|K Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. Here's how they described it: Equations commonly used in Discrete Math. Cheat Sheet /MediaBox [0 0 612 792] Learn everything from how to sign up for free to enterprise WebBefore tackling questions like these, let's look at the basics of counting. Web2362 Education Cheat Sheets. Question A boy lives at X and wants to go to School at Z. Web445 Cheatsheet. \newcommand{\st}{:} << Once we can count, we can determine the likelihood of a particular even and we can estimate how long a Counting 69 5.1. $A \cap B = \emptyset$), 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. Then, The binomial expansion using Combinatorial symbols. Cheatsheet - Summary Discrete Mathematics I stream No. ]\}$ be such that for all $i$, $A_i\neq\varnothing$. ];_. 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. There are $50/6 = 8$ numbers which are multiples of both 2 and 3. 3 0 obj Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. + \frac{ (n-1)! } /Length 58 '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}
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