\newcommand{\card}[1]{\left| #1 \right|} Cardinality of power set is , where n is the number of elements in a set. 9 years ago Then, The binomial expansion using Combinatorial symbols. Here's how they described it: Equations commonly used in Discrete Math. Types of propositions based on Truth values1.Tautology A proposition which is always true, is called a tautology.2.Contradiction A proposition which is always false, is called a contradiction.3.Contingency A proposition that is neither a tautology nor a contradiction is called a contingency. 1 0 obj << \newcommand{\Q}{\mathbb Q} The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! Math/CS cheat sheet. Learn more. WebBefore tackling questions like these, let's look at the basics of counting. '1g[bXlF) q^|W*BmHYGd tK5A+(R%9;P@2[P9?j28C=r[%\%U08$@`TaqlfEYCfj8Zx!`,O%L v+ ]F$Dx U. 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. %PDF-1.4 Download the PDF version here. on April 20, 2023, 5:30 PM EDT. << 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! Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. /Width 156 One of the first things you learn in mathematics is how to count. 17 0 obj WebLets dene the positive integers using the set builder notation: N+= {x : x N and x > 0}. \newcommand{\inv}{^{-1}} 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? See Last Minute Notes on all subjects here. Discrete case Here, $X$ takes discrete values, such as outcomes of coin flips. Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, element of the domain. No. + \frac{ n-k } { k!(n-k)! } \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. ];_. 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. To guarantee that a graph with n vertices is connected, minimum no. Assume that s is not 0. Hence, there are (n-2) ways to fill up the third place. \renewcommand{\bar}{\overline} xWn7Wgv /ca 1.0 stream stream /ProcSet [ /PDF ] 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 These are my notes created after giving the same lesson 4-5 times in one week. There must be at least two people in a class of 30 whose names start with the same alphabet. <> endobj %PDF-1.2 The permutation will be $= 6! Probability 78 Chapter 7. ]8$_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 Representations of Graphs 88 7.3. WebThe first principle of counting involves the student using a list of words to count in a repeatable order. CS160 - Fall Semester 2015. +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. 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 Thus, n2 is odd. Thereafter, he can go Y to Z in $4 + 5 = 9$ ways (Rule of Sum). of relations =2mn7. 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. :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. WebDiscrete Math Cram Sheet alltootechnical.tk 7.2 Binomial Coefcients The binomial coefcient (n k) can be dened as the co-efcient of the xk term in the polynomial *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! (c) Express P(k + 1). stream Combinatorics 71 5.3. In other words a Permutation is an ordered Combination of elements. Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). 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. 5 0 obj There are $50/6 = 8$ numbers which are multiples of both 2 and 3. Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). 6 0 obj set of the common element in A and B. DisjointTwo sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements. Share it with us! \newcommand{\va}[1]{\vtx{above}{#1}} xKs6. 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! /ProcSet [ /PDF /Text ] BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. \newcommand{\imp}{\rightarrow} Proof : Assume that m and n are both squares. From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. Graph Theory 82 7.1. /Filter /FlateDecode (b) Express P(k). >> endobj Did you make this project? Note that in this case it is written \mid in LaTeX, and not with the symbol |.
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