A professor in a discrete mathematics class passes out a form asking students to check all the mathematics and computer science courses they have recently taken. The finding is that out of a total of 50 students in the class, 30 took precalculus; 16 took both precalculus and Java; 18 took calculus; 8 took both calculus and Java; 26 took Java; 47 took at least one of the three courses; and 9 took both precalculus and calculus. a. How many students did not take any of the three courses? b. How many students took all three courses? c. How many students took precalculus and calculus but not Java? How many students took precalculus but neither calculus nor Java?

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Let the $J, C, P$ be the sets of those that took Java, Calculus and Precalculus courses, respectively.

It is know that the overall amount of students is 50.

the amount of students in $P = 30$

the amount of students in $J\bigcap P = 16$

the amount of students in $C = 18$

the amount of students in $J\bigcap C = 8$

the amount of students in $J = 26$

the amount of students in $J\bigcup C \bigcup P = 47$

the amount of students in $C\bigcap P = 9$

A. Since the are 50 students and 47 took at least one course, there are $50 -47 = 3$ such students

B. $J\bigcup C \bigcup P = J + C + P +$

$+(- J\bigcap C - J\bigcap P -C\bigcap P + J\bigcap C \bigcap P) =$

$= 26 + 18 + 30 - 8 - 16 - 9 + J\bigcap C \bigcap P =$

$= 41 + J\bigcap C \bigcap P$

$47 = 41 + J\bigcap C \bigcap P$

$J\bigcap C \bigcap P = 6$

6 students took both Java, Calculus and Precalculus courses.

C. $(C\bigcap P)\setminus J = C\bigcap P - C\bigcap P \bigcap J = 9 - 6 = 3$

3 students took Precalculus and Calculus but not Java.

$P\setminus (C\bigcup J) = P - P\bigcap(C\bigcup J)=$

$=P - (P\bigcap C)\bigcup (P\bigcap J).$

$(P\bigcap C)\bigcup (P\bigcap J) =$

$= P\bigcap C + P\bigcap J -(P\bigcap C)\bigcap (P\bigcap J) =$

$= P\bigcap C + P\bigcap J -P\bigcap C\bigcap J =$

$=$ 9 + 16 - 6 = 19

$P - (P\bigcap C)\bigcup (P\bigcap J) = 30 - 19 = 11 =$

$=P\setminus (C\bigcup J)$

11 students took precalculus but neither calculus nor Java

Answer:

a. 3 students did not take any of the three courses.

b. 6 students took both Java, Calculus and Precalculus courses.

c. 3 students took Precalculus and Calculus but not Java.

11 students took precalculus but neither calculus nor Java.