A professor has 24 text books on computer science and is concerned about their coverage of the topics (P) compilers, (Q) data structures and (R) Operating systems. The following data gives the number of books that contain material on these topics: n(P) = 8, n(Q) = 13, n(R) = 13, n(P ∩ R) = 3, n(P ∩ R) = 3, n(Q ∩ R) = 3, n(Q ∩ R) = 6, n(P ∩ Q ∩ R) = 2, where n(x) is the cardinality of the set x. Then the number of text books that have no material on compilers is
16
Understanding Textbook Coverage Analysis
This problem involves analyzing the coverage of topics in a collection of computer science textbooks using principles of set theory. We are given the total number of textbooks and the number of books covering specific topics: compilers (P), data structures (Q), and operating systems (R). We are also provided with information about the overlaps between these topics.
Identifying Key Information
Let's list the crucial information provided:
- Total number of textbooks = 24.
- Number of books covering compilers, denoted as $n(P) = 8$.
- Number of books covering data structures, denoted as $n(Q) = 13$.
- Number of books covering operating systems, denoted as $n(R) = 13$.
- Number of books covering compilers and operating systems, $n(P \cap R) = 3$.
- Number of books covering data structures and operating systems, $n(Q \cap R) = 6$.
- Number of books covering all three topics (compilers, data structures, and operating systems), $n(P \cap Q \cap R) = 2$.
The question specifically asks for the number of textbooks that have no material on compilers.
Calculating Books Without Compiler Material
The phrase "no material on compilers" means we are looking for textbooks that are outside the set P. In set theory terms, this is the complement of set P, denoted as $P^c$. The number of elements in the complement of a set within a universal set is found by subtracting the number of elements in the set from the total number of elements in the universal set.
The formula to find the number of textbooks with no material on compilers is:
$$ n(P^c) = \text{Total Number of Textbooks} - n(P) $$Applying the Formula
We substitute the given values into the formula:
- Total Number of Textbooks = 24
- Number of books covering compilers, $n(P) = 8$
Calculation:
$$ n(P^c) = 24 - 8 $$ $$ n(P^c) = 16 $$Conclusion on Textbook Coverage
Therefore, there are 16 textbooks that have no material on compilers. It is important to note that the information regarding data structures (Q), operating systems (R), and the intersections ($P \cap R$, $Q \cap R$, $P \cap Q \cap R$) was not necessary to answer this specific question about compilers.
Important Questions from Operations on Sets
Match List I with List II
Let R 1= {(1, 1), (2, 2), (3, 3)} and R 2 = {(1, 1), (1, 2), (1, 3), (1, 4)}
List I
List II
(A) R 1∪ R 2
(I) {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (3, 3)}
(B) R 1- R 2
(II) {1, 1}
(C) R 1∩ R 2
(III) {(1, 2), (1, 3), (1, 4)}
(D) R 2- R 1
(IV) {(2, 2), (3, 3)}
Choose the correct answer from the options given below:
Let R be a relation on a set A such that R = R-1, then R is
Which of the following is an open set?
In a beauty contest, half the number of experts voted for Mr. A and two third voted for Mr. B. 10 voted for both and 6 did not for either. How many experts were there in all?
In a survey where 100 students reported which subjects they like, 32 students in total liked Mathematics, 38 students liked Business and 30 students liked Literature. Moreover 7 students liked both Mathematics and Literature, 10 students liked both Mathematics and Business, 8 students liked both Business and Literature, 5 students liked all three subjects.
Then the number of people who liked exactly one subject is
