Sets, Set Operations, and Venn Diagrams

Given the universal set $U = \{x \in \mathbb{Z} : 1 \le x \le 20\}$. Let $P = \{x \in U : x \text{ is a prime number}\}$, $Q = \{x \in U : x \text{ is an odd integer}\}$, and $R = \{x \in U : x \text{ is a multiple of } 3\}$. What is the value of $n((P \cup Q) \cap R')$?

In a survey of $100$ agricultural market traders in Lagos, $52$ sell cassava, $45$ sell yam, and $60$ sell plantain. Furthermore, $25$ sell both cassava and yam, $22$ sell both yam and plantain, and $28$ sell both cassava and plantain. If the number of traders who sell none of these three crops is twice the number of traders who sell all three crops, how many traders sell exactly two of these crops?

In a class of $40$ students, $25$ study Mathematics and $18$ study Physics. If $3$ students study neither of the two subjects, how many students study both Mathematics and Physics?

Let the universal set be $U = \{x \in \mathbb{Z} : 1 \le x \le 10\}$. If $P = \{x : x \text{ is a prime number within } U\}$ and $Q = \{x : x \text{ is an even number within } U\}$, which of the following represents the set $(P \cup Q)'$?

In a survey of $150$ subscribers of a digital media platform, $75$ prefer High-Definition Audio ($H$), $70$ prefer Offline Downloads ($D$), and $65$ prefer Ad-free Listening ($A$). It is observed that $35$ subscribers prefer both $H$ and $D$, $30$ prefer both $D$ and $A$, and $25$ prefer both $H$ and $A$. If $15$ subscribers prefer none of these three features, how many subscribers prefer exactly two of these features?

Given the universal set $U = \{x \in \mathbb{Z} : 1 \le x \le 15\}$, let $A = \{x \in U : x \text{ is a multiple of } 3\}$ and $B = \{x \in U : x \text{ is an even number}\}$. What is the cardinality of $(A \cup B)'$?

In a sports academy of $85$ athletes, $52$ participate in track events, $43$ participate in field events, and $12$ participate in neither track nor field events. How many athletes participate in both track and field events?

In an agricultural survey of $100$ local farmers in a community, $55$ grow Cassava ($C$), $45$ grow Yam ($Y$), and $40$ grow Maize ($M$). It is observed that $20$ farmers grow both Cassava and Yam, $15$ grow both Yam and Maize, and $16$ grow both Cassava and Maize. If $7$ farmers grow all three crops, how many farmers grow none of these three crops?

In a medical study of $120$ hospital outpatients, $68$ are prescribed Green tea extract ($G$), $52$ are prescribed Herbal infusion ($H$), and $44$ are prescribed Chamomile syrup ($C$). Furthermore, $24$ patients receive both Green tea extract and Herbal infusion, $18$ receive both Green tea extract and Chamomile syrup, $16$ receive both Herbal infusion and Chamomile syrup, while $8$ receive all three prescriptions. How many patients receive at most one of these three prescriptions?

In a technology conference of $60$ software developers, $35$ write code in Python, $28$ write in Java, and $12$ write in both Python and Java. How many developers write in neither of these two languages?

In a department of $70$ university lecturers, $40$ publish research in Journal $A$, $30$ publish in Journal $B$, and $25$ publish in Journal $C$. It is known that $15$ publish in both Journals $A$ and $B$, $12$ publish in both Journals $B$ and $C$, $10$ publish in both Journals $A$ and $C$, and $5$ publish in all three journals. How many lecturers do not publish in any of these three journals?

In a survey of $120$ town residents regarding three newspapers—*The Herald* ($H$), *The Express* ($E$), and *The Nation* ($N$)—it was found that $52$ read *The Herald*, $45$ read *The Express*, and $60$ read *The Nation*. Furthermore, $15$ read both *The Herald* and *The Express*, $22$ read both *The Express* and *The Nation*, and $18$ read both *The Herald* and *The Nation*. If the number of residents who read none of these three newspapers is twice the number of those who read all three newspapers, how many residents read exactly two of the newspapers?

On an international flight carrying $150$ passengers, each passenger was offered three meal options: Chicken ($C$), Fish ($F$), and Vegetarian ($V$). A survey of their choices showed that $75$ passengers chose Chicken, $60$ chose Fish, and $50$ chose Vegetarian. Additionally, $15$ passengers chose both Chicken and Fish, $12$ chose both Fish and Vegetarian, $18$ chose both Chicken and Vegetarian, while $8$ passengers chose all three meals. How many passengers chose none of the three meal options?

Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and a subset $P = \{x \in U : x \text{ is a prime number}\}$, which of the following sets represents the complement of $P$, denoted as $P'$?

In a survey of $200$ agricultural exporters regarding three major commodities—Cocoa ($C$), Palm Oil ($P$), and Rubber ($R$)—it was found that $110$ export Cocoa, $90$ export Palm Oil, and $75$ export Rubber. Exactly $20$ exporters export none of the three commodities, and $45$ export Cocoa only. Furthermore, the number of exporters who export Cocoa and Palm Oil only is twice the number of exporters who export all three commodities; the number who export Palm Oil and Rubber only is equal to the number who export all three; and the number who export Cocoa and Rubber only is $5$ more than the number who export all three. Find the total number of exporters who export at least two of the three commodities.

In a survey of $180$ cloud computing engineers regarding their proficiency in three major platforms—AWS ($A$), Azure ($B$), and Google Cloud ($C$)—it was found that $95$ are proficient in AWS, $80$ in Azure, and $75$ in Google Cloud. Furthermore, $40$ are proficient in both AWS and Azure, $35$ in both Azure and Google Cloud, $30$ in both AWS and Google Cloud, and $15$ are not proficient in any of the three platforms. How many engineers are proficient in exactly one of these platforms?

In a survey of $120$ tech entrepreneurs at an innovation hub, $65$ secured Angel Investment ($A$), $55$ received Venture Capital ($V$), and $50$ obtained Government Grants ($G$). Furthermore, $25$ received both $A$ and $V$, $20$ received both $V$ and $G$, $22$ received both $A$ and $G$, while $12$ received no funding from any of these three sources. How many entrepreneurs secured funding from exactly two of these sources?