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Chapter 7: Problem 26

Let \(U\) denote the set of all students who applied for admission to the freshman class at Faber College for the upcoming academic year, and let \(A=\\{x \in U \mid x\) is a successful applicant \(\\}\) \(B=\\{x \in U \mid x\) is a female student who enrolled in the freshman class\\} \(C=\\{x \in U \mid x\) is a male student who enrolled in the freshman class\\} a. Use Venn diagrams to represent the sets \(U, A, B\), and \(C\). b. Determine whether the following statements are true or false. i. \(A \subseteq B\) ii. \(B \subset A\) iii. \(C \subset B\)

Short Answer

Expert verified
In short, the Venn diagram represents Sets U, A, B, and C, where Set A includes both male and female students who enrolled, Set B includes female students, and Set C includes male students. Analyzing the statements: i. \(A \subseteq B\) is false; ii. \(B \subset A\) is true; iii. \(C \subset B\) is false.

Step by step solution

01

Understanding the Sets

First, understand each set as follows: - Set U = All students who applied for admission to the freshman class - Set A = Successful applicants (students admitted) - Set B = Female students who enrolled in the freshman class - Set C = Male students who enrolled in the freshman class
02

Draw Venn Diagrams for Sets U, A, B, and C

Draw a Venn Diagram that includes 4 circles for Sets U, A, B, and C, respectively. - Outer rectangle represents Set U - Circle A represents Set A (successful applicants) - Circle B represents Set B (female students who enrolled) - Circle C represents Set C (male students who enrolled) Next, overlap Circle A with circles B and C. This represents that students from Sets B and C are a part of Set A (successful applicants).
03

Analyze Statements

Now, analyze each statement on the Venn diagram created in Step 2. i. \(A \subseteq B\): Is A a subset of B (Are all successful applicants also female students who enrolled)? False. Set A includes both male and female students who enrolled, not only the female students. ii. \(B \subset A\): Is B a proper subset of A (Is any female student who enrolled also a successful applicant)? True. All students in Set B are included in Set A (as they have to be successful applicants to enroll), but Set A also has other students (males) not in Set B. iii. \(C \subset B\): Is C a proper subset of B (Are all male students who enrolled also female students)? False. Set C includes male students who enrolled while Set B includes female students, so these sets are separate and not a subset of each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Venn diagrams
Understanding complex set relationships can often be challenging, but Venn diagrams are a powerful visual tool to simplify the process. Venn diagrams use overlapping circles to represent different sets and their relationships with one another. Each circle contains elements that belong to a particular set, and the areas where circles overlap represent common elements shared between sets.

For the exercise involving sets of students applying to Faber College, Venn diagrams enable us to depict the relationship between all applicants (set U), successful applicants (set A), and those who enrolled classified by gender (sets B and C). The outer rectangle encompasses the universal set U, while the circles for A, B, and C illustrate successful applicants and enrolled students, with B and C being segregated by gender and nested within A to show that enrollment is contingent upon being a successful applicant.

Through Venn diagrams, students can visually assess the correctness of statements regarding subsets and proper subsets, which is crucial for a deeper comprehension of set logic in real-world contexts like college admissions.
Subset and Proper Subset
The concept of a subset is foundational in set theory. A subset is a set where all elements of one set are also elements of another. For example, if set A is a subset of set U, it means every member of A is also a member of U, symbolically written as \(A \subseteq U\).

A proper subset takes this concept a step further. If set B is a proper subset of set A, denoted as \(B \subset A\), then every element of B is also in A, but set A contains at least one element not found in B. This is equivalent to saying set B is contained within set A but does not completely represent set A.

In our college application example, all female students who enrolled (set B) are successful applicants (set A), making B a proper subset of A. However, not all successful applicants are female students who enrolled; therefore, A is not a subset of B. Recognizing whether a set is a subset or proper subset is essential in analyzing set relations and solving related problems effectively.
Set Notation
Set notation is the standardized language of set theory, allowing mathematicians and students to express complex set relationships succinctly and precisely. This symbolic language includes brackets for listing elements, and special symbols, such as \(\subseteq\), \(\subset\), and \(\in\), that denote subset relationships and membership, respectively.

In set notation, curly braces denote the elements of a set, for instance, \(A = \{x \in U \mid x \) is a successful applicant \(\}\). This reads as set A consists of elements x such that x belongs to the universal set U and meets the condition of being a successful applicant. Similarly, the symbols for subset (\(\subseteq\)) and proper subset (\(\subset\)) are critical for expressing the hierarchy and exclusivity of sets in relation to one another.

Understanding and using set notation correctly is vital for communicating effectively in mathematics, especially when dealing with large or abstract sets. Mastery of this notation enables students to interpret and solve problems with clarity and confidence.

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Most popular questions from this chapter

Suppose the probability that Bill can solve a problem is \(p_{1}\) and the probability that Mike can solve it is \(p_{2}\). Show that the probability that Bill and Mike working independently can solve the problem is \(p_{1}+p_{2}-p_{1} p_{2}\).

According to a study conducted in 2003 concerning the participation, by age, of \(401(\mathrm{k})\) investors, the following data were obtained: $$ \begin{array}{lccccc} \hline \text { Age } & 20 \mathrm{~s} & 30 \mathrm{~s} & 40 \mathrm{~s} & 50 \mathrm{~s} & 60 \mathrm{~s} \\ \hline \text { Percent } & 11 & 28 & 32 & 22 & 7 \\ \hline \end{array} $$ a. What is the probability that a \(401(\mathrm{k})\) investor selected at random is in his or her 20 s or 60 s? b. What is the probability that a \(401(\mathrm{k})\) investor selected at random is under the age of 50 ?

The probability that a shopper in a certain boutique will buy a blouse is .35, that she will buy a pair of pants is \(.30\) and that she will buy a skirt is \(.27\). The probability that she will buy both a blouse and a skirt is \(.15\), that she will buy both a skirt and a pair of pants is \(.19\), and that she will buy both a blouse and a pair of pants is \(.12\). Finally, the probability that she will buy all three items is .08. What is the probability that a customer will buy a. Exactly one of these items? b. None of these items?

In an online survey for Talbots of 1095 women ages \(35 \mathrm{yr}\) and older, the participants were asked what article of clothing women most want to fit perfectly. A summary of the results of the survey follows: $$ \begin{array}{lc} \hline \text { Article of Clothing } & \text { Respondents } \\ \hline \text { Jeans } & 470 \\ \hline \text { Black Pantsuit } & 307 \\ \hline \text { Cocktail Dress } & 230 \\ \hline \text { White Shirt } & 22 \\ \hline \text { Gown } & 11 \\ \hline \text { Other } & 55 \\ \hline \end{array} $$ If a woman who participated in the survey is chosen at random, what is the probability that she most wants a. Jeans to fit perfectly? b. A black pantsuit or a cocktail dress to fit perfectly?

In a television game show, the winner is asked to select three prizes from five different prizes, \(A, B\), \(\mathrm{C}, \mathrm{D}\), and \(\mathrm{E} .\) a. Describe a sample space of possible outcomes (order is not important). b. How many points are there in the sample space corresponding to a selection that includes A? c. How many points are there in the sample space corresponding to a selection that includes \(\mathrm{A}\) and \(\mathrm{B}\) ? d. How many points are there in the sample space corresponding to a selection that includes either \(\mathrm{A}\) or \(\mathrm{B}\) ?
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