Set Theory and Venn Diagrams: A Teacher's Approach to Solving Overlapping Sets Problems

Recently, I encountered a homework problem from a student's class packet: 'In a class of 35 students, 23 took science and 18 took art, 11 students took both science and art. How many did not take either science or art?'

While it might be tempting to jump straight to a solution, this problem presents an excellent opportunity to explore set theory and Venn diagrams in a more engaging and instructive manner. This method can help students grasp a foundational concept that applies to many other areas of mathematics and problem-solving in general.

Understanding the Problem

At first glance, this may seem like a straightforward arithmetic problem. However, it can quickly become confusing if we don’t use a structured approach. The keyword here is ‘overlapping sets’—specifically, students who are taking both subjects are counted in both groups, leading to a situation where the total number of students taking at least one subject is not simply the sum of the two groups.

Using Venn Diagrams

To solve this problem effectively, the first step is to visualize it using a Venn diagram. This graphical representation of sets makes it easier to understand the relationships and overlaps between different categories.

Let’s start by creating the Venn diagram:

The Venn Diagram

We have:

The entire class of 35 students as the outer circle. A yellow oval representing the set of students who took science. A blue oval representing the set of students who took art. A green section representing the overlap, or students who took both subjects.

This visual representation helps us break down the problem step by step:

Breaking Down the Sets

1. The number of students who only took science is 23 - 11 12.

2. The number of students who only took art is 18 - 11 7.

3. The number of students who took both subjects is 11, as given.

4. Adding these up, we get the total number of students who took either science or art:

Students who only took science: 12 Students who only took art: 7 Students who took both: 11 Total: 12 7 11 30

This sum represents those who took either science or art, but we need to consider the students who didn’t take any of these subjects.

The All-Encompassing Solution

Since there are 35 students in total and 30 of them took either science or art, we can find the number of students who didn’t take either subject by subtracting the number of students who took either subject from the total class size:

Total students: 35 Students taking either science or art: 30 Students taking neither: 35 - 30 5

Therefore, the answer to the problem is 5 students.

Conclusion

This approach not only helps in solving the problem but also teaches the importance of visual representation and step-by-step problem-solving strategies. By using Venn diagrams and set theory, we can systematically break down complex problems into manageable parts, ensuring a clear understanding of the underlying concepts.

While the teacher's style might not always match a student's learning style, it’s important to remember that every teaching approach can serve as a valuable learning tool. Engaging with the problem through different methods can enhance comprehension and retention.

Feel free to practice more over time to solidify your understanding and apply these techniques to new, more challenging problems.