Understanding 'Or' in Set Theory: Solving the Math and Science Pupil Problem

The question often presents students with confusing data: 'In a class of 60 pupils, 30 like math, 25 like science, and 25 like both math and science. How many pupils like the two subjects?'

This puzzle has sparked debate with multiple interpretations, leading to seemingly contradictory answers. Let's delve into the nuances of the question and explore how to solve it accurately.

Breaking Down the Problem

Let's define variables for clarity. Let m be the number of students who like math and s be the number of students who like science. We are given:
- 30 students like math (m)
- 25 students like science (s)
- 25 students like both math and science (overlap)

Using Set Theory to Solve

When solving such problems, it is crucial to use the principles of set theory, particularly Venn diagrams. A Venn diagram can help visualize the relationships between the sets. Let's break down the problem into four segments:

Students who like only math: This is m - overlap Students who like only science: This is s - overlap Students who like both math and science: This is the given overlap (25) Students who like neither subject

Using the values provided:

Students who like only math: 30 - 25 5 Students who like only science: 25 - 25 0 Students who like both math and science: 25

Now, to find the total number of students who like either math or science or both, we sum up the three segments:

5 (only math) 0 (only science) 25 (both) 30

Interpreting the 'Or' in the Question

The question can be interpreted in two ways due to the use of 'or':

Exclusive OR: Students who answer 'or' mean those who like only one subject, excluding those who like both. This is the sum of 'only math' and 'only science' Inclusive OR: Students who answer 'or' mean those who like either one subject or both, including those who like both. This is the sum of 'only math', 'only science', and 'both'

Inclusive OR:

5 (only math) 25 (only science) 25 (both) 55

Exclusive OR:

5 (only math) 25 (only science) 30

Visualizing with a Venn Diagram

A Venn diagram can clarify the problem visually. Draw an outer rectangle to represent the 60 students in the class. Inside this rectangle, draw two overlapping circles:

One circle for math, labeled 30 total, with 15 in the overlap area One circle for science, labeled 25 total, with 15 in the overlap area

The remaining area outside these circles but within the rectangle represents students who like neither subject. To find this, we subtract the sum of the three segments from the total number of students:

60 - (5 25 25) 10 students who like neither subject

Conclusion

The puzzle is resolved by understanding the two interpretations of 'or': exclusive and inclusive. The correct answer depends on the specific meaning intended by the question. A thorough understanding of set theory and visual aids like Venn diagrams can provide clear solutions.