Question

(5)(3 pts) There are 16 students in a class. Twelve of those students are taking a language class, either French, English or Spanish. Seven of the students are taking French, eight are taking English and seven are taking Spanish. Three students are taking French and Spanish, five are taking Spanish and English and four are taking French and English. Use the Principle of Inclusion-Exclusion to find the number of students taking all three courses. Let F, E and S represent the sets of students taking French, English and Spanish respectively. Show Draw a Venn diagram and find the cardinality of the following sets. set of students taking either French or English or both set of students taking either English or Spanish or both set of students taking either French or Spanish or both set of students taking none of these classes set of students taking exactly one of these classes set of students taking exactly two of these classes

Answer 1

Solution- from the above universal diagram We know n (EUFUS) = 12 n(u) = 16 nCENFOS) = ? → hCF) = 7 n (E) - 8 nas) = 7 nens) - 3 nc S9E) - 5 nC EOE) = 4 We know that nCEUFUS) = n(E) + ACF) + n(s) - n cins)-n(SE)+-n(Fe)+ nc Eoins) nCEDENS) = nCEUFUS) - ACE) - (F)- n(s) + n(FOS) +n(SNE) + OCENE) = = = = 12-8-7-7+3+5+4 12-8-7-7+8+4 12-14+4 12-10 = ?

VELLE @ no of Students taking either French or english or bol DCEUF) = nCE) +nCF)- nCEOF) nCE) -8 n() = 7 nCEOF)=4 = 8+7-4 ir bolt o no of students taking either english or Spanish or bot ACEUS) = n(E)+ncs) - nCens) nCE) = 8 = 8+7-5 n05) = 7 = 0 nCEns)=5 of students taking either French or Spanish or € No both n(SUF) = n(s)+nCF) - 5 (SOF) = 7+7-3 = 14-3 nas) = 7 n(a)=7 n(595)=3 © Number of students none Seilffects = n(el) - nC EUFUS) = 16-12 nu=16 na EU FUS)-2 wo of students taking exa therefore = No of students wo of students wo of students only take French + only fake Englisht only take spanish

= nCF) at - n( Fns)-ACENE) +nCEnens)+ InCE) - nCEns) - ACFOE) + n(Foens)) + In (5) - nesne) - n(SOF) +nCENENS J = 7+847 - [7-3-4 +2] + 8-5-4 +2] + [7-5-372) = [2+1+1] © No of Students taking exactly 2 classes INFOE)- nCEnens Incens) - nCFnens)] + Incens)-n(Foens)] [4-2] + [5-23 +13-2]] = [2+3+13