i) If X,Y,ZX,Y,Z are finite sets, then |X∩Y∩Zc|=|?1∩Y|−|X∩Y∩Z|where ?1=_______(ii) If X,Y,Z are finite sets, then |X∩Yc∩Zc|=|X|−|X∩?2∩Zc|−|X∩?3∩Z|−|X∩Y∩Z|where ?2=___________ where ?3=___________ (iii) A certain department of mathematics offers three elective courses: a course in abstract algebra (A), a course in mathematical biology (B), and a course in computer science (C). It is known that in a group of 41 students of the department: 29 students took the course in abstract algebra (A), 23 students took the course in mathematical biology (B), and 21 students took the course in computer science (C).12 students took both (A) and (C), 16 took (A) and (B), 13 took (B) and (C);8 took all three courses.Write A (resp. B,C) for the set of students who took the course (A) (resp. (B), (C)).(a) Use (i) to find the cardinalities of the fundamental products A∩B∩Cc,A∩Bc∩C,Ac∩B∩C,of the sets A,B,C:|A∩B∩Cc|=_______________|A∩Bc∩C|=_______________|Ac∩B∩C|=_______________(b) Use (iii) to find the cardinalities of the fundamental products A∩Bc∩Cc,Ac∩B∩Cc,Ac∩Bc∩Cof the sets A,B,C:|A∩Bc∩Cc|=_______________|Ac∩B∩Cc|=_______________|Ac∩Bc∩C|=_______________(c) Find the cardinality of the fundamental product Ac∩Bc∩CcAc∩Bc∩Cc:|Ac∩Bc∩Cc|=_______________(d) Use (a,b,c) to find the number of students that took:- only (A):_____________- only (B):_____________- only (C):_____________- (B) and (C) but not (A):_____________- (A) and (B) but not (C): _____________- only one of the courses:_____________- at least one course:_____________- none of the courses: Answer_____________ PLEASE ANSWER ALL PART OF THE QUESTION

if X , Y , Z are finite sets (i) X∩Y∩ZC=?∩Y-X∩Y∩Z it is known that X∩Y∩ZC=X∩Y-X∩Y∩Z

Answered: i) If X,Y,ZX,Y,Z are finite sets, then…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.1: Sets

Problem 41E: Express (AB)(AB) in terms of unions and intersections that involve A,A,B,andB

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Question

i) If X,Y,ZX,Y,Z are finite sets, then

|X∩Y∩Zc|=|?1∩Y|−|X∩Y∩Z|

where ?₁=_______

(ii) If X,Y,Z are finite sets, then

|X∩Yc∩Zc|=|X|−|X∩?₂∩Zc|−|X∩?₃∩Z|−|X∩Y∩Z|

where ?_{_{2=___________}}where ?_{3=___________}

_{(iii) A certain department of mathematics offers three elective courses: a course in abstract algebra (A), a course in mathematical biology (B), and a course in computer science (C). It is known that in a group of 41 students of the department:}

_{29 students took the course in abstract algebra (A), 23 students took the course in mathematical biology (B), and 21 students took the course in computer science (C).}
_{12 students took both (A) and (C), 16 took (A) and (B), 13 took (B) and (C);}
_{8 took all three courses.}

_{Write A (resp. B,C) for the set of students who took the course (A) (resp. (B), (C)).}

(a) Use (i) to find the cardinalities of the fundamental products

A∩B∩C^c,A∩B^c∩C,A^c∩B∩C,

of the sets A,B,C:

|A∩B∩C^c|=_______________

|A∩B^c∩C|=_______________

|A^c∩B∩C|=_______________

(b) Use (iii) to find the cardinalities of the fundamental products

A∩B^c∩C^c,A^c∩B∩C^c,A^c∩B^c∩C

of the sets A,B,C:

|A∩B^c∩C^c|=_______________

|A^c∩B∩C^c|=_______________

|A^c∩B^c∩C|=_______________

|A^c∩B^c∩C^c|=_______________

(d) Use (a,b,c) to find the number of students that took:

- only (A):_____________

- only (B):_____________

- only (C):_____________

- (B) and (C) but not (A):_____________

- (A) and (B) but not (C): _____________

- only one of the courses:_____________

- at least one course:_____________

- none of the courses: Answer_____________

PLEASE ANSWER ALL PART OF THE QUESTION

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