WebDefinitions. A graph is euler graph if it there exists atmost 2 vertices of odd degree9. endobj on April 20, 2023, 5:30 PM EDT. stream \newcommand{\lt}{<} \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. <> ]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 /Filter /FlateDecode Hence, there are (n-1) ways to fill up the second place. Problem 3 In how ways can the letters of the word 'ORANGE' be arranged so that the consonants occupy only the even positions? WebI COUNTING Counting things is a central problem in Discrete Mathematics. Proof Let there be n different elements. Note that in this case it is written \mid in LaTeX, and not with the symbol |. By using our site, you Question A boy lives at X and wants to go to School at Z. Then(a+b)modm= ((amodm) + 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-1.5 /CreationDate (D:20151115165753Z) )$. 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. Combinatorics 71 5.3. (\frac{ k } { k!(n-k)! } >> Thus, n2 is odd. /Parent 22 0 R Hi matt392, nice work! endobj \newcommand{\U}{\mathcal U} SA+9)UI)bwKJGJ-4D tFX9LQ (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d >> endobj 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. \renewcommand{\iff}{\leftrightarrow} | x | = { x if x 0 x if x < 0. There are two very important equivalences involving quantifiers. 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 $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. I have a class in it right now actually! %PDF-1.3 No. \definecolor{fillinmathshade}{gray}{0.9} /SMask /None>> 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 $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. Simple is harder to achieve. Prove or disprove the following two statements. Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. /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 WebChapter 5. Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. Prove the following using a proof by contrapositive: Let x be a rational number. Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! `y98R uA>?2 AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! stream [Q hm*q*E9urWYN#-&\" e1cU3D).C5Q7p66[XlG|;xvvANUr_B(mVt2pzbShb5[Tv!k":,7a) Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. NOTE: Order of elements of a set doesnt matter. \newcommand{\card}[1]{\left| #1 \right|} No. Definitions // Set A contains elements 1,2 and 3 A = {1,2,3} WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. The cardinality of the set is 6 and we have to choose 3 elements from the set. For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$. \renewcommand{\v}{\vtx{above}{}} 23 0 obj << *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! xm=j0 gRR*9BGRGF. \newcommand{\isom}{\cong} Discrete Mathematics - Counting Theory - TutorialsPoint 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 $P(a < X\leqslant B)=F(b)-F(a)$. 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. of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. /Filter /FlateDecode Size of the set S is known as Cardinality number, denoted as |S|. Probability For Dummies Cheat Sheet - dummies Cheat Sheet of Mathemtical Notation and Terminology 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 From a night class at Fordham University, NYC, Fall, 2008. ]$, The number of circular permutations of n different elements taken x elements at time = $^np_{x}/x$, The number of circular permutations of n different things = $^np_{n}/n$. $A \cap B = \emptyset$), 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. = 720$. I hate discrete math because its hard for me to understand. We have: Chebyshev's inequality Let $X$ be a random variable with expected value $\mu$. Now we want to count large collections of things quickly and precisely. 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. Once we can count, we can determine the likelihood of a particular even and we can estimate how long a 14 0 obj '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} 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 on April 20, 2023, 5:30 PM EDT. \newcommand{\C}{\mathbb C} 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. Counting mainly encompasses fundamental counting rule, the permutation rule, and the combination rule. Course Hero is not sponsored or endorsed by any college or university. 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. Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional In 1834, German mathematician, Peter Gustav Lejeune Dirichlet, stated a principle which he called the drawer principle. Get up and running with ChatGPT with this comprehensive cheat sheet. WebBefore tackling questions like these, let's look at the basics of counting. $c62MC*u+Z Variance The variance of a random variable, often noted Var$(X)$ or $\sigma^2$, is a measure of the spread of its distribution function. *3-d[\HxSi9KpOOHNn uiKa, 1 This is a matter of taste. Let q = a b and r = c d be two rational numbers written in lowest terms. 6 0 obj >> English to French cheat sheet, with useful words and phrases to take with you on holiday. It includes the enumeration or counting of objects having certain properties. U denotes the universal set. 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. /Subtype /Image Here it means the absolute value of x, ie. So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. Mathematically, if a task B arrives after a task A, then $|A \times B| = |A|\times|B|$. We make use of First and third party cookies to improve our user experience. /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 >> How to Build a Montessori Bookshelf With Just 2 Plywood Sheets. \newcommand{\vr}[1]{\vtx{right}{#1}} I go out of my way to simplify subjects. Minimum number of connected components =, 6. Discrete Math 1: Set Theory. Cheat Sheet | by Alex Roan - Medium To guarantee that a graph with n vertices is connected, minimum no. cheat sheet >> 5 0 obj Assume that s is not 0. Equal setsTwo sets are said to be equal if both have same elements. \newcommand{\Z}{\mathbb Z} Prove that if xy is irrational, then y is irrational. I'll check out your sheet when I get to my computer. Math Counting - Discrete Mathematics 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 Hence, a+c b+d(modm)andac bd(modm). of edges in a complete graph = n(n-1)/22. element of the domain. /Width 156 /Length 530 \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} Above Venn Diagram shows that A is a subset of B. DMo`6X\uJ.~{y-eUo=}CLU6$Pendstream (nr+1)! By using this website, you agree with our Cookies Policy. (1!)(1!)(2!)] /Parent 22 0 R /Type /Page You can use all your notes, calcu-lator, and any books you Sum of degree of all vertices is equal to twice the number of edges.4. \newcommand{\st}{:} A country has two political parties, the Demonstrators and the Repudiators. Discrete Mathematics of irreflexive relations = 2n(n-1), 15. Hence, the number of subsets will be $^6C_{3} = 20$. /Resources 1 0 R 5 0 obj 25 0 obj << Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, <> Toomey.org Tutoring Resources In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. endobj 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. \newcommand{\Iff}{\Leftrightarrow} % It is determined as follows: Characteristic function A characteristic function $\psi(\omega)$ is derived from a probability density function $f(x)$ and is defined as: Euler's formula For $\theta \in \mathbb{R}$, the Euler formula is the name given to the identity: Revisiting the $k^{th}$ moment The $k^{th}$ moment can also be computed with the characteristic function as follows: Transformation of random variables Let the variables $X$ and $Y$ be linked by some function. endobj xY8_1ow>;|D@`a%e9l96=u=uQ Solution There are 6 letters word (2 E, 1 A, 1D and 2R.) Tree, 10. That Vertical bar sign in Discrete mathematics Find the number of subsets of the set $\lbrace1, 2, 3, 4, 5, 6\rbrace$ having 3 elements. stream Minimum no. >> endobj For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. 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. Bnis the set of binary strings with n bits. WebIB S level Mathematics IA 2021 Harmonics and how music and math are related. Paths and Circuits 91 3 /Type /ExtGState The Rule of Sum and Rule of Product are used to decompose difficult counting problems into simple problems. Generalized Permutations and Combinations 73 5.4. of edges to have connected graph with n vertices = n-17. /ca 1.0 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, of edges =m*n3. xKs6. Web445 Cheatsheet. /Filter /FlateDecode /\: [(2!) 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 \newcommand{\gt}{>} Then m 2n 4. Besides, your proof of 0!=1 needs some more attention. No. Every element has exactly one complement.19. In this case the sign means that a divides b, or that b a is an integer. Graphs 82 7.2. 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! This ordered or stable list of counting words must be at least as long as the number of items to be counted. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . 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 $c$. No. << The function is injective (one-to-one) if every element of the codomain is mapped to by at most one. WebBefore tackling questions like these, let's look at the basics of counting. A relation is an equivalence if, 1. No. of onto function =nm (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m . /Type /Page For solving these problems, mathematical theory of counting are used. @>%c0xC8a%k,s;b !AID/~ /Height 25 /Resources 23 0 R 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. /Filter /FlateDecode on April 20, 2023, 5:30 PM EDT. \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} % 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. 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. 4 0 obj We say that $\{A_i\}$ is a partition if we have: Remark: for any event $B$ in the sample space, we have $\displaystyle P(B)=\sum_{i=1}^nP(B|A_i)P(A_i)$. of symmetric relations = 2n(n+1)/29. Different three digit numbers will be formed when we arrange the digits. Let s = q + r and s = e f be written in lowest terms. \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. (c) Express P(k + 1). /First 812 DISCRETE MATHEMATICS FOR COMPUTER SCIENCE 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. If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. The permutation will be $= 6! \). A combination is selection of some given elements in which order does not matter. CPS102 DISCRETE MATHEMATICS Practice Final Exam - Duke (nr+1)!$, The number of permutations of n dissimilar elements when r specified things never come together is $n![r! Then, The binomial expansion using Combinatorial symbols. \PAwX:8>~\}j5w}_rP*%j3lp*j%Ghu}gh.~9~\~~m9>U9}9 Y~UXSE uQGgQe 9Wr\Gux[Eul\? Then, number of permutations of these n objects is = $n! \(\renewcommand{\d}{\displaystyle} He may go X to Y by either 3 bus routes or 2 train routes. The no. Then n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. If we consider two tasks A and B which are disjoint (i.e. endobj of edges required = {(n-1)*(n-2)/2 } + 18. 9 years ago of reflexive relations =2n(n-1)8. 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 Cheatsheet v1.1.1 Simpsons Paradox Expected The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! /Creator () +(-1)m*(n, C, n-1), if m >= n; 0 otherwise4. { r!(n-r)! /Filter /FlateDecode Get up and running with ChatGPT with this comprehensive cheat sheet. xS@}WD"f<7.\$.iH(Rc'vbo*g1@9@I4_ F2 }3^C2>2B@>8JfWkn%;?t!yb C;.AIyir!zZn}Na;$t"2b {HEx}]Zg;'B!e>3B=DWw,qS9\ THi_WI04$-1cb }$, $= (n-1)! Distributive Lattice : Every Element has zero or 1 complement .18. 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 $S = \lbrace 1, 2, 3 \rbrace$. FWfSE xpwy8+3o After filling the first place (n-1) number of elements is left. 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. Math/CS cheat sheet. The Pigeonhole Principle 77 Chapter 6. ~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. in the word 'READER'. Discrete Math Review 1 0 obj << Necessary condition for bijective function |A| = |B|5. ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream xWn7Wgv Before tackling questions like these, let's look at the basics of counting. /Contents 25 0 R c o m) x3T0 BCKs=S\.t;!THcYYX endstream For complete graph the no . The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. /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 . WebThe Discrete Math Cheat Sheet was released by Dois on Cheatography. 3 0 obj << WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. % Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide 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 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 $P(n,r)\geqslant C(n,r)$. 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! 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. \dots (a_r!)]$. No. 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 of asymmetric relations = 3n(n-1)/211. Discrete Math 1: Set Theory Cheat Sheet Photo by Gabby K from Pexels (not actually discrete math) 1. So an enthusiast can read, with a title, short definition and then formula & transposition, then repeat. mathematics ChatGPT cheat sheet: Complete guide for 2023 >> So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. 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. 1 0 obj stream 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. \newcommand{\vl}[1]{\vtx{left}{#1}} How many ways are there to go from X to Z? 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. Then m 3n 6. 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. In this case it is written with just the | symbol. Discrete Math Cheat Sheet by Dois #education #mathematics #math 9 years ago 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. Complemented Lattice : Every element has complement17. See Last Minute Notes on all subjects here. 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 $w_1 + w_2 + \dots +w_m$. << of one to one function = (n, P, m)3. 6 0 obj In other words a Permutation is an ordered Combination of elements. Mathematics | Combinatorics Basics /AIS false Counting 69 5.1.

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