Question

3. Three components are connected to form a system as shown in the accompanying diagram. Because the compo- nents in the 2–3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1 must function and so must the 2–3 subsystem. 214 The experiment consists of determining the condition of each component [S (success) for a functioning compo- nent and F (failure) for a nonfunctioning component]. a. Which outcomes are contained in the event A that exactly two out of the three components function? b. Which outcomes are contained in the event B that at least two of the components function? c. Which outcomes are contained in the event C that the system functions? d. List outcomes in C', AUC, Anc, BUC, and Bnc. 9. Use Venn diagrams to verify the following two relationships for any events A and B (these are called De Morgan's laws): a. (A U B)' = A' N B' b. (AN B)' = A' U B' [Hint: In each part, draw a diagram corresponding to the left side and another corresponding to the right side.]

Answer 1

1.3. Solution: (aj to list the outcomes in A, write down all combination in which & components are functioning E r is non functioning. This consists of a all 3 combinations where I of the letters are s & letter inf o A = {SSF,SFS, FSS} The outcomes with a funckoning & functoning component comisk of AEE SSF, SAS, FSS non- ibe The oukame where are least o component que functioning fall into o Categories Categoly. nis that {nachly o components aic functioning. ccitegoly teso that all 3 components que functioning. * The list outcomes is then all outcomes in both categories Category 'i' was determined in partea). Add the outcome where all 3" components are functioning to part cas B = {SSF, SFS, FSS, SSS} * The outcomes with cutleast o funckoning component que bu{SSF, SF1,FA, ...SSSg . "" In order to the system to function. component i must fonction and at least one o component . and 3 must function. Weiling down the posible outcomes (-{SSF, SFS, SSS} * The outcomes where the syskon funcions que c= {SSF, $F5, SSS 3

Lab The complement '; the event that the system does not function. an doel for the system not to function, components must not work for both hion. While down there posible outcomes c': {Fs, FSF, FFS, EFF, SFF3 * The outcomes where the system does not funckion are ca { fss, FSF, FAS, FFF, SFF * The outcomes in Auc are all outcomes that appeal in A&c. using pouts a Ec, combine the outcomes from sek AEC AUC - {SSF, SF), F5SJ :{sSF, SAJ, S153 3 . SSF., SFS, FJS, SSS} : :: The outcomes in Auc ale Auc='ASSF, SFI, FJS, SSSÝ * The outcomes in Anc are all outcomes that appear in both A&C .. using parts a&c. find the outcomes shared by seh A&C Anc = {SSF, SA, fss} n {SS F, SFS, SSS} l b- - {SSF, SF53. ..The outcomes in Anc aue Anc = {SSF, SESİ * The outcomes in BUC are all outcomes that appear in Boc. - using parts bec. combine the outcomes from sét B&C. Notice that ic is contained in B. so their union will be B. Buc = {sst, SF, FSS, ssc? o'{SSF, SA, SSS} : { SSF, IFS, FJS ,5853 The outcomes in Bnc cue all outcomes that appeal in both B&C using pauls bac. And the outcomes shared by ret B&c. Nohile that can contained in B, so their intersection will bec.

BAC = {SSF, SFS, ASS, SSS n {SSF, SÉS, Susy - {SSF, SFS, siss} Theis the outcomes is Bnc ale BAC = {JSF, SFS, CSSÝ events A&B (a) consides the following relationship between (AUB)'-A'ns! relationship į shown in figuee. AUB venn diagram for (Alue)!

venn drageam Je A' venndiagram Ja B' Therefore the venn diagram for AnB' is - venn diagram for Ans' ibi consider the following relationship behueen Events A & B CAOB)'-A'ud' U venn diagram jos Ang..

Venn diagram for (ANB)' venn diagram for AUB .: The shaded regions for both Events (ADB) & Aub' cue identical So the eelationship Hold good