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So n(S∪B)=100 , n(S)=80 and n(B)=70 We want to know n(S∩B) which is equal to n(S)n(B)−n(S∪B)=8070−100=50 by the second form of the general addition rule Venn Diagrams A Venn diagram can be a useful way of depicting sets and set operations We use circles to represent the sets, and enclose our diagram in a rectangle" _ " `Note If U is a universal set and A is a subset of U,thenn(Ac)=n(U)n(A) 1 Let the universal set U = {1,2,3,4,5,6,7,8,9,10} Find the following (a) n(U) (b) n(Ac), where A = {x xisanevennumberfrom1to10} (c) n(B), where B = {1,3,9} (d) n(?) Addition Rule for Sets Very Useful Formula If A and B are finite sets then n(AB)=n(A)n(B)n(A\B) 2

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ƒuƒƒbƒN 40 'ã 'j« "¯Œ^- BPUNJ No 13 GAS FiftySeventh Revised Sheet No 46 EMERGENCY SALES SERVICE CHARGE APPLICABLE TO RATE SCHEDULES RSG, GSG, LVG, SLG, TSGF, TSGNF (Per Therm) Public Service's BGSS supplier(s)'s weighted average pipeline transportation costClearly this derivation (and some variations on it) produce bmabnabp for each m, n, p 2 (a) G = (V, Σ, R, S) with V = {S, a, b, c}, Σ = {a, b, c}, R = { S → aSa S → bSb S → c } (b) Same as (a) except remove c from V and Σ and replace the last rule, S → c, by S → ε (c) This language very similar to the language of (b) (b) was

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, A 3 0 are thirty sets each with five elements and B 1 , B 2 ,, B n are n sets each with three elements such that ⋃ i = 1 3 0 A i = ⋃ j = 1 n B j = S and each element of S belongs to exactly 1 0 of the A i 's and exactly 9 of the B j 's Then n =(b) Let U be uniformly distributed on (0,1) Determine an increasing function g on (0,1) such that g(U) has the same distribution as Y (c) Determine constants a and b > 0 such that the random variable a bY has lower quartile 0 and upper quartile 1 (d) Determine the variance of the random variable abY, where a and b areB ⋅ Energy of E The energy of an electrostatic configuration U = 1 1 2 2 ∫ V ρφdV = 8π∫ E dV Pressure A layer of surface charge density σ exerts a pressure P = 2πσ 2 G G Current density J =ρv GG Current I = dQ dt / = ∫J da ⋅ ( I is the current through surface S ) S GG∂ρ Continuity ∇⋅J

) " e i e v u o f t s o l j e r e q g i e p o n m l k j g i h g f e d c x w s o l l y u f l y u f x w s o l e i e v u o f t s o l j e r e q g i i e j h e z l k j g i h g f e d c # ; Definition binomial A binomial is an algebraic expression containing 2 terms For example, (x y) is a binomial We sometimes need to expand binomials as follows (a b) 0 = 1(a b) 1 = a b(a b) 2 = a 2 2ab b 2(a b) 3 = a 3 3a 2 b 3ab 2 b 3(a b) 4 = a 4 4a 3 b 6a 2 b 2 4ab 3 b 4(a b) 5 = a 5 5a 4 b 10a 3 b 2 10a 2 b 3 5ab 4 b 5Clearly,By the authority vested in me as President by the Constitution and the laws of the United States of America, including the Federal Property and Administrative Services Act of 1949, 40 USC 101 et seq, and in order to take further steps to promote economy and efficiency in Federal Government procurement, it is hereby ordered as follows

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N=1 and (b n) 1 n=1 represent sequences of real numbers For each statement, decide if it is true or false If it is true, give a proof If it is false, provide a counterexample (a) If (a n) and (b n) are both bounded, then so is (a n b n) Solution This is true, since if ja nj Mand jb nj M0for all n, then by the triangle inequality, ja n< ^ \ !MAT 111 – Practice Chapter 2 Test 1 Express the set using set – builder notation {spring, summer, winter, fall} {x x is a season of the year}

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Created Date 5/7/12 AMPubMed® comprises more than 33 million citations for biomedical literature from MEDLINE, life science journals, and online books Citations may include links to full text content from PubMed Central and publisher web sites24 c JFessler,May27,04,1310(studentversion) 212 Classication of discretetime signals The energy of a discretetime signal is dened as Ex 4= X1 n=1 jxnj2 The average power of a signal is dened as Px 4= lim N!1 1 2N 1 XN n= N jxnj2 If E is nite (E < 1) then xn is called an energy signal and P = 0 If E is innite, then P can be either nite or innite

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Venn Diagrams We can visual subsets of a universal set, and how they interact/overlap, using Venn diagrams, as shown below On the left, the brown shaded region is A\BExample 3 (Continued) Find the dot product of u and v u v = a1 a2 b1 b2 u v = (6)(3) (2)(5) u v = 18 – 10 u v = 8 Find the angle between the vectors cos 1 uv uv θ − ⋅ = () cos 1 8 210 34 θ= − θ≈cos −1 θ≈° 775Official MapQuest website, find driving directions, maps, live traffic updates and road conditions Find nearby businesses, restaurants and hotels Explore!

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