Discrete Mathematics (1) If X, Y, Z are finite sets, then|XNYN Z°| = |Xn1- |XNYNZ|,where ?1(i) If X, Y, Z are finite sets, then|XnY°n Z°| = |X| – |Xn 2n z°| – |Xn23 nZ| – |XnYn ZI,where ?2and ?3-(ii) 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 computerscience (C). It is known that in a group of 43 students of the department:• 24 students took the course in abstract algebra (A), 16 students took the course in mathematical biology (B), and 26 students took the course in computer science (C);• 16 students took both (A) and (C), 10 took (A) and (B), 10 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 () to find the cardinalities of the fundamental productsAn.C°, An B" nC, A°nBnc,of the sets A, B, C:|ANBNC"| =|ANBºNC| =- - (b) Use (ii) to find the cardinalities of the fundamental productsAnB nc", A° n Bnc", A°n B° ncof the sets A, B, C:|AN Ben c°| =|A°nBnc| =-|A°n B° nC[ =(c) Find the cardinality of the fundamental product A“ n BºnCe:(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:

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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

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Discrete Mathematics

(1) If X, Y, Z are finite sets, then
|XNYN Z°| = |Xn1- |XNYNZ|,
where ?1
(i) If X, Y, Z are finite sets, then
|XnY°n Z°| = |X| – |Xn 2n z°| – |Xn23 nZ| – |XnYn ZI,
where ?2
and ?3
-
(ii) 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 43 students of the department:
• 24 students took the course in abstract algebra (A), 16 students took the course in mathematical biology (B), and 26 students took the course in computer science (C);
• 16 students took both (A) and (C), 10 took (A) and (B), 10 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 () to find the cardinalities of the fundamental products
An.
C°, An B" nC, A°nBnc,
of the sets A, B, C:
|ANBNC"| =
|ANBºNC| =
- -

(b) Use (ii) to find the cardinalities of the fundamental products
AnB nc", A° n Bnc", A°n B° nc
of the sets A, B, C:
|AN Ben c°| =
|A°nBnc| =
-
|A°n B° nC[ =
(c) Find the cardinality of the fundamental product A“ n BºnCe:
(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:

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