Let G be a connected planar simple graph with n vertices, where n ? 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. (b) Express P(k). stream Event Any subset $E$ of the sample space is known as an event. 3 0 obj << 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. Problem 1 From a bunch of 6 different cards, how many ways we can permute it? Web2362 Education Cheat Sheets. Every element has exactly one complement.19. Generalized Permutations and Combinations 73 5.4. After filling the first place (n-1) number of elements is left. \(\renewcommand{\d}{\displaystyle} 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 Pascal's Identity. xm=j0 gRR*9BGRGF. From a set S ={x, y, z} by taking two at a time, all permutations are , We have to form a permutation of three digit numbers from a set of numbers $S = \lbrace 1, 2, 3 \rbrace$. /Length 1781 Define the set Ento be the set of binary strings with n bits that have an even number of 1's. In how many ways we can choose 3 men and 2 women from the room? For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? I have a class in it right now actually! Size of a SetSize of a set can be finite or infinite. >> c o m) Complemented Lattice : Every element has complement17. /SM 0.02 Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. >> endobj 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID =HA`miNDKH;8&.2_LcVfgsIVAxx$A,t([k9QR$jmOX#Q=s'0z>SUxH-5OPuVq+"a;F} WebProof : Assume that n is an odd integer. 3 0 obj Let G be a connected planar simple graph with n vertices and m edges, and no triangles. o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X CS160 - Fall Semester 2015. \). 9 years ago +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. @ys(5u$E$VY(@[Y+J(or(0ze7+s([nlY+J(or(0zemFGn2+%f mEH(X (nr+1)! WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. Probability 78 6.1. Web445 Cheatsheet. 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 endobj A permutation is an arrangement of some elements in which order matters. WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. of the domain. Hi matt392, nice work! English to French cheat sheet, with useful words and phrases to take with you on holiday. (1!)(1!)(2!)] + \frac{ (n-1)! } of spanning tree possible = nn-2. 25 0 obj << Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. of edges in a complete graph = n(n-1)/22. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. >> Cartesian ProductsLet A and B be two sets. ~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. The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! 14 0 obj \renewcommand{\iff}{\leftrightarrow} xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d Did you make this project? How many different 10 lettered PAN numbers can be generated such that the first five letters are capital alphabets, the next four are digits and the last is again a capital letter. xS@}WD"f<7.\$.iH(Rc'vbo*g1@9@I4_ F2 }3^C2>2B@>8JfWkn%;?t!yb C;.AIyir!zZn}Na;$t"2b {HEx}]Zg;'B!e>3B=DWw,qS9\ THi_WI04$-1cb /Type /ObjStm Show that if m and n are both square numbers, then m n is also a square number. A combination is selection of some given elements in which order does not matter. /Subtype /Image There are n number of ways to fill up the first place. \newcommand{\st}{:} WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. /Length 530 ];_. We can also write N+= {x N : x > 0}. Below is a quick refresher on some math tools and problem-solving techniques from 240 (or other prereqs) that well assume knowledge of for the PSets. Pascal's identity, first derived by Blaise Pascal in 17th century, states that the number of ways to choose k elements from n elements is equal to the summation of number of ways to choose (k-1) elements from (n-1) elements and the number of ways to choose elements from n-1 elements. Then(a+b)modm= ((amodm) + of bijection function =n!6. By using this website, you agree with our Cookies Policy. << a b. WebLets dene the positive integers using the set builder notation: N+= {x : x N and x > 0}. ]\}$ be such that for all $i$, $A_i\neq\varnothing$. <> on April 20, 2023, 5:30 PM EDT. Math/CS cheat sheet. x3T0 BCKs=S\.t;!THcYYX endstream 1 This is a matter of taste. 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. Learn more. >> stream of asymmetric relations = 3n(n-1)/211. 1 0 obj << Cumulative distribution function (CDF) The cumulative distribution function $F$, which is monotonically non-decreasing and is such that $\underset{x\rightarrow-\infty}{\textrm{lim}}F(x)=0$ and $\underset{x\rightarrow+\infty}{\textrm{lim}}F(x)=1$, is defined as: Remark: we have $P(a < X\leqslant B)=F(b)-F(a)$. % \newcommand{\gt}{>} I go out of my way to simplify subjects. NOTE: Order of elements of a set doesnt matter. 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 of one to one function = (n, P, m)3. See Last Minute Notes on all subjects here. 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. 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 3n 6. We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. 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. 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 . 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. Probability 78 Chapter 7. 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. 9 years ago 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. In complete bipartite graph no. The no. \newcommand{\vr}[1]{\vtx{right}{#1}} of symmetric relations = 2n(n+1)/29. 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. /SA true Here's how they described it: Equations commonly used in Discrete Math. Note that in this case it is written \mid in LaTeX, and not with the symbol |. \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} SA+9)UI)bwKJGJ-4D tFX9LQ = 180.$. In this case the sign means that a divides b, or that b a is an integer. Affordable solution to train a team and make them project ready. 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. Graphs 82 7.2. In general, use the form It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. 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 /Decode [1 0] \newcommand{\U}{\mathcal U} That's a good collection you've got there, but your typesetting is aweful, I could help you with that. Rsolution chap02 - Corrig du chapitre 2 de benson Physique 2; CCNA 1 v7 Modules 16 17 Building and Securing a Small Network Exam Answers; Processing and value addition in ornamental flower crops (2019-AJ-66) Chapitre 3 r ponses (STE) Homework 9.3 endobj /Type /Page `y98R uA>?2 AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. &IP")0 QlaK5 )CPq 9n TVd,L j' )3 O@ 3+$ >+:>Ov?! For solving these problems, mathematical theory of counting are used. endobj >> endobj of relations =2mn7. = 720$. 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 $(a_1 + a_2 + a_r) = n$. There are 6 men and 5 women in a room. Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. Minimum no. \newcommand{\Z}{\mathbb Z} No. Let s = q + r and s = e f be written in lowest terms. You can use all your notes, calcu-lator, and any books you /Filter /FlateDecode For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. 5 0 obj << 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. Representations of Graphs 88 7.3. No. 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? Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures.