Problem 37
Question
According to a survey among 160 college students, 95 students take a course in English, 72 take a course in French, 67 take a course in German, 35 take a course in English and in French, 37 take a course in French and in German, 40 take a course in German and in English, and 25 take a course in all three languages. Find the number of students in the survey who take a course in: Neither English, French, nor German.
Step-by-Step Solution
Verified Answer
The number of students in the survey who take a course in neither English, French, nor German is \(13\).
1Step 1: Determine the total number of students taking each language
We are given the number of students taking English, French, and German:
- 95 students take a course in English
- 72 students take a course in French
- 67 students take a course in German
2Step 2: Determine the number of students taking two languages at once
We are given the number of students taking two languages at once:
- 35 students take a course in English and in French
- 37 students take a course in French and in German
- 40 students take a course in German and in English
3Step 3: Determine the number of students taking all three languages
We are given the number of students taking all three languages:
- 25 students take a course in English, French, and German
4Step 4: Apply the principle of inclusion-exclusion
Using the principle of inclusion-exclusion, we can break down the number of students taking any of the three languages as:
Number of students taking any language = Total number of students taking each language - Total number of students taking two languages at once + Number of students taking all three languages
We will plug in the numbers from Steps 1, 2, and 3:
Number of students taking any language = (95 + 72 + 67) - (35 + 37 + 40) + 25
Number of students taking any language = (234) - (112) + 25 = 122 + 25 = 147
5Step 5: Find the number of students taking neither English, French, nor German
Now, to find the number of students not taking any of the three languages, we will subtract the number of students taking any language (which we found in Step 4) from the total number of students surveyed (160):
Number of students taking neither English, French, nor German = Total number of students surveyed - Number of students taking any language
Number of students taking neither English, French, nor German = 160 - 147 = 13
So, there are 13 students in the survey who take none of the languages: English, French, or German.
Key Concepts
Set TheorySurveys in MathematicsDiscrete Mathematics
Set Theory
At its core, set theory is a branch of mathematical logic that studies collections of objects, known as sets. It's foundational for various areas of mathematics and offers a way to understand how groups of items relate to each other.
Consider our problem with students and language courses. Each course—English, French, and German—can be thought of as a set consisting of students. When we discuss students taking multiple languages, we're looking at the intersection of these sets, or the number of students who fall into multiple groups simultaneously. However, the complexity surfaces in accounting for students counted multiple times because they're in multiple sets. Here, set theory presents the principle of inclusion-exclusion, a critical concept to avoid overcounting. It ensures that when we calculate the total number of unique students, we include them once no matter how many sets they belong to.
The textbook exercise utilizes this principle by treating courses as sets and students as elements. We include the number of students in individual courses (sets), exclude those in the intersections of two courses (overlapping sets), and then make sure to include those who are in the intersection of all three sets since they've been excluded twice.
Consider our problem with students and language courses. Each course—English, French, and German—can be thought of as a set consisting of students. When we discuss students taking multiple languages, we're looking at the intersection of these sets, or the number of students who fall into multiple groups simultaneously. However, the complexity surfaces in accounting for students counted multiple times because they're in multiple sets. Here, set theory presents the principle of inclusion-exclusion, a critical concept to avoid overcounting. It ensures that when we calculate the total number of unique students, we include them once no matter how many sets they belong to.
The textbook exercise utilizes this principle by treating courses as sets and students as elements. We include the number of students in individual courses (sets), exclude those in the intersections of two courses (overlapping sets), and then make sure to include those who are in the intersection of all three sets since they've been excluded twice.
Surveys in Mathematics
In the realm of mathematics, surveys pertain to the methodical collection and analysis of data sets to study frequency, distribution, and relationships between variables—in this case, course enrollment. Surveys in education often help draw significant conclusions about student behavior or preferences.
In our exercise, a survey among college students is used to gather data regarding language course enrollments. The raw data is a starting point. Though we have the numbers, interpreting what they mean for the subject at hand is where the mathematical analysis comes in. Through the use of set theory and the principle of inclusion-exclusion, we translate the survey data into a coherent answer to a simple question: How many students are not enrolled in any language course? This application of theory to interpret surveys is a common practice in discrete mathematics, as it involves finite collections of data points and allows for precise, unequivocal answers.
In our exercise, a survey among college students is used to gather data regarding language course enrollments. The raw data is a starting point. Though we have the numbers, interpreting what they mean for the subject at hand is where the mathematical analysis comes in. Through the use of set theory and the principle of inclusion-exclusion, we translate the survey data into a coherent answer to a simple question: How many students are not enrolled in any language course? This application of theory to interpret surveys is a common practice in discrete mathematics, as it involves finite collections of data points and allows for precise, unequivocal answers.
Discrete Mathematics
Discrete mathematics is the study of structures that are fundamentally discrete rather than continuous. It includes topics such as logic, set theory, graph theory, and combinatorics. Discreteness means that there is a distinct, countable separation between values, which fits perfectly with our exercise—every student can be counted and separated into different course sets.
The principle of inclusion-exclusion is vital in discrete mathematics because it's a way of counting that ensures accuracy where multiple criteria overlap. It's an elegant solution to potentially complex counting problems found in surveys in mathematics and beyond. Importantly, this concept is applicable not just in theory but in practical uses such as computer science, database structuring, and even epidemiology.
Understanding this principle can massively improve a student's ability to handle various problems within discrete mathematics. By learning to apply this approach to set-related problems, students will be equipped with crucial tools for handling real-world data analysis situations.
The principle of inclusion-exclusion is vital in discrete mathematics because it's a way of counting that ensures accuracy where multiple criteria overlap. It's an elegant solution to potentially complex counting problems found in surveys in mathematics and beyond. Importantly, this concept is applicable not just in theory but in practical uses such as computer science, database structuring, and even epidemiology.
Understanding this principle can massively improve a student's ability to handle various problems within discrete mathematics. By learning to apply this approach to set-related problems, students will be equipped with crucial tools for handling real-world data analysis situations.
Other exercises in this chapter
Problem 36
\(n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2 \text { and } 3\\}\)
View solution Problem 37
In Exercises \(34-37, n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2 \text
View solution Problem 37
Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$A \subseteq A \cap B$$
View solution Problem 37
\(n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2 \text { or } 3\\}\)
View solution