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 That's a good collection you've got there, but your typesetting is aweful, I could help you with that. 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 A country has two political parties, the Demonstrators and the Repudiators. Then, number of permutations of these n objects is = $n! stream 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 The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. Hence, there are (n-1) ways to fill up the second place. /Length 1781 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. 14 0 obj The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! )$. *3-d[\HxSi9KpOOHNn uiKa, Representations of Graphs 88 7.3. /ca 1.0 In this case the sign means that a divides b, or that b a is an integer. 1 0 obj << So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. This implies that there is some integer k such that n = 2k + 1. 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. Note that zero is an even number, so a string. 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 $w_1 + w_2 + \dots +w_m$. From there, he can either choose 4 bus routes or 5 train routes to reach Z. 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. Graphs 82 7.2. 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. %PDF-1.2 If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. Proof : Assume that m and n are both squares. { (k-1)!(n-k)! } @ys(5u$E$VY(@[Y+J(or(0ze7+s([nlY+J(or(0zemFGn2+%f mEH(X on April 20, 2023, 5:30 PM EDT. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. How many ways can you distribute \(10\) girl scout cookies to \(7\) boy scouts? DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream of Anti Symmetric Relations = 2n*3n(n-1)/210. \newcommand{\va}[1]{\vtx{above}{#1}} /Contents 3 0 R Here it means the absolute value of x, ie. In general, use the form \renewcommand{\bar}{\overline} By using our site, you 28 0 obj << \newcommand{\inv}{^{-1}} 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. In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. /Resources 23 0 R of edges required = {(n-1)*(n-2)/2 } + 18. Once we can count, we can determine the likelihood of a particular even and we can estimate how long a Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. \dots (a_r!)]$. >> After filling the first and second place, (n-2) number of elements is left. Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. How many integers from 1 to 50 are multiples of 2 or 3 but not both? WebI COUNTING Counting things is a central problem in 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. 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. xmT;s1Wli+,[-:^Q1GL$E=>]KC}{~=ogwh=9-} }pNY@z }>c? From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. \renewcommand{\v}{\vtx{above}{}} For complete graph the no . o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X 2195 ~C'ZOdA3,3FHaD%B,e@,*/x}9Scv\`{]SL*|)B(u9V|My\4 Xm$qg3~Fq&M?D'Clk +&$.U;n8FHCfQd!gzMv94NU'M`cU6{@zxG,,?F,}I+52XbQN0.''f>:Vn(g."]^{\p5,`"zI%nO. Here's how they described it: Equations commonly used in Discrete Math. 6 0 obj No. E(aX+bY+c) =aE(X) +bE(Y) +c If two Random Variables have the same distribution, even when theyare dependent by theproperty of Symmetrytheir expected = 6$ ways. Show that if m and n are both square numbers, then m n is also a square number. >> Here, the ordering does not matter. Did you make this project? stream Besides, your proof of 0!=1 needs some more attention. 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 $P(n,r)\geqslant C(n,r)$. Cardinality of power set is , where n is the number of elements in a set. One of the first things you learn in mathematics is how to count. ]\}$ be a partition of the sample space. /MediaBox [0 0 612 792] stream WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. Minimum no. No. Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$. Size of a SetSize of a set can be finite or infinite. /Contents 25 0 R \newcommand{\pow}{\mathcal P} Suppose that the national senate consists of 100 members, 44 of which are Demonstrators and 56 of which are Rupudiators. Above Venn Diagram shows that A is a subset of B. }$, $= (n-1)! The function is surjective (onto) if every element of the codomain is mapped to by at least one element. Define the set Ento be the set of binary strings with n bits that have an even number of 1's. /Length 530 Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. No. = 180.$. endobj ];_. \(\renewcommand{\d}{\displaystyle} 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. xm=j0 gRR*9BGRGF. Event Any subset $E$ of the sample space is known as an event. 9 years ago stream a b. of bijection function =n!6. 5 0 obj << /SA true n Less theory, more problem solving, focuses on exam problems, use as study sheet! /ProcSet [ /PDF /Text ] \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} >> WebProof : Assume that n is an odd integer. % Let q = a b and r = c d be two rational numbers written in lowest terms. of reflexive relations =2n(n-1)8. If we consider two tasks A and B which are disjoint (i.e. /Width 156 Learn more. \newcommand{\gt}{>} Mathematically, for any positive integers k and n: $^nC_{k} = ^n{^-}^1C_{k-1} + ^n{^-}^1{C_k}$, $= \frac{ (n-1)! } \newcommand{\R}{\mathbb R} English to French cheat sheet, with useful words and phrases to take with you on holiday. /Length 58 Complemented Lattice : Every element has complement17. 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. There must be at least two people in a class of 30 whose names start with the same alphabet. Probability 78 6.1. xY8_1ow>;|D@`a%e9l96=u=uQ &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 stream (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. /Producer ( w k h t m l t o p d f) We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. stream Download the PDF version here. Hence, the total number of permutation is $6 \times 6 = 36$. stream Size of the set S is known as Cardinality number, denoted as |S|. Corollary Let m be a positive integer and let a and b be integers. Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional Before tackling questions like these, let's look at the basics of counting. 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 Agree No. \newcommand{\lt}{<} /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 . /N 100 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 \newcommand{\imp}{\rightarrow} Probability 78 Chapter 7. /First 812 on April 20, 2023, 5:30 PM EDT. The cardinality of the set is 6 and we have to choose 3 elements from the set. ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream <> Solution There are 6 letters word (2 E, 1 A, 1D and 2R.) Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). %PDF-1.4 In other words a Permutation is an ordered Combination of elements. Every element has exactly one complement.19. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! I'll check out your sheet when I get to my computer. 25 0 obj << Share it with us! x[yhuv*Nff&oepDV_~jyL?wi8:HFp6p|haN3~&/v3Nxf(bI0D0(54t,q(o2f:Ng #dC'~846]ui=o~{nW] Math/CS cheat sheet. Then, The binomial expansion using Combinatorial symbols. on Introduction. Necessary condition for bijective function |A| = |B|5. /Height 25 Now, it is known as the pigeonhole principle. FWfSE xpwy8+3o So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. | x | = { x if x 0 x if x < 0. \newcommand{\amp}{&} It includes the enumeration or counting of objects having certain properties. To guarantee that a graph with n vertices is connected, minimum no. + \frac{ n-k } { k!(n-k)! } Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. endobj 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. Hence, the number of subsets will be $^6C_{3} = 20$. this looks promising :), Reply Note that in this case it is written \mid in LaTeX, and not with the symbol |. WebBefore tackling questions like these, let's look at the basics of counting. How many ways are there to go from X to Z? How many like both coffee and tea? Graph Theory 82 7.1. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe 9Wr\Gux[Eul\? Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. /Filter /FlateDecode Tree, 10. Different three digit numbers will be formed when we arrange the digits. % \newcommand{\st}{:} Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. BKT~1ny]gOzQzErRH5y7$a#I@q\)Q%@'s?. The permutation will be $= 6! Sum of degree of all vertices is equal to twice the number of edges.4. Paths and Circuits 91 3 Helps to encode it into the brain. Minimum number of connected components =, 6. 9 years ago 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. in the word 'READER'. \newcommand{\vb}[1]{\vtx{below}{#1}} Web445 Cheatsheet. I go out of my way to simplify subjects. /MediaBox [0 0 612 792] No. There are $50/3 = 16$ numbers which are multiples of 3. Generalized Permutations and Combinations 73 5.4. The function is injective (one-to-one) if every element of the codomain is mapped to by at most one. A permutation is an arrangement of some elements in which order matters. Bnis the set of binary strings with n bits. Binomial Coecients 75 5.5. This ordered or stable list of counting words must be at least as long as the number of items to be counted. From his home X he has to first reach Y and then Y to Z. <> /Filter /FlateDecode Learn everything from how to sign up for free to enterprise Solution From X to Y, he can go in $3 + 2 = 5$ ways (Rule of Sum). :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx How many anagrams are there of anagram? Now we want to count large collections of things quickly and precisely. 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. 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, 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? \newcommand{\Q}{\mathbb Q} $c62MC*u+Z /CA 1.0 3 and m edges. \newcommand{\Z}{\mathbb Z} /Type /XObject 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. Simple is harder to achieve. /Font << /F17 6 0 R /F18 9 0 R /F15 12 0 R /F7 15 0 R /F8 18 0 R /F37 21 0 R >> A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. >> /Creator () (1!)(1!)(2!)] 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. Thank you - hope it helps. <> No. endobj Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? \newcommand{\Iff}{\Leftrightarrow} $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. 23 0 obj << c o m) By noting $f_X$ and $f_Y$ the distribution function of $X$ and $Y$ respectively, we have: Leibniz integral rule Let $g$ be a function of $x$ and potentially $c$, and $a, b$ boundaries that may depend on $c$. = 6$. A combination is selection of some given elements in which order does not matter. U denotes the universal set. 17 0 obj of relations =2mn7. Then m 3n 6. That is, an event is a set consisting of possible outcomes of the experiment. { k!(n-k-1)! 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. Thus, n2 is odd. WebBefore tackling questions like these, let's look at the basics of counting. 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 , $|A \cup B \cup C | = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C |$, $|\bigcup_{i=1}^{n}A_i|=\sum\limits_{1\leq i0$, 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. | x |. Remark 2: If X and Y are independent, then $\rho_{XY} = 0$. WebLets dene the positive integers using the set builder notation: N+= {x : x N and x > 0}. >> endobj We can also write N+= {x N : x > 0}. Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. 6 0 obj Cartesian ProductsLet A and B be two sets. \newcommand{\isom}{\cong} (nr+1)! 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 connected components in graph with n vertices = n5. 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 $n_{P_{r}}$. I hate discrete math because its hard for me to understand. /SM 0.02 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. Let G be a connected planar simple graph with n vertices and m edges, and no triangles. WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. Question A boy lives at X and wants to go to School at Z. From a night class at Fordham University, NYC, Fall, 2008. Define P(n) to be the assertion that: j=1nj2=n(n+1)(2n+1)6 (a) Verify that P(3) is true. << We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. Basic rules to master beginner French! }28U*~5} Kryi1#8VVN]dVOJGl\+rlN|~x lsxLw:j(b"&3X]>*~RrKa!

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