Solving Combining Football and Cricket Players Using Set Theory

In a class of 75 students, 35 play football, 20 play both football and cricket, while the rest play cricket. Let's apply the principles of set theory to determine how many students play only football, only cricket, or both.

Understanding the Problem

We are given the following information:

Total number of students 75 Students who play football (F) 35 Students who play both football and cricket (F ∩ C) 20

We need to find:

The number of students who play only football (F - F ∩ C) The number of students who play only cricket (C - F ∩ C)

Step-by-Step Solution

Step 1: Calculate the number of students who play only football

To find the number of students who play only football, we can use the formula:

F only F - F ∩ C

Given:

F 35 F ∩ C 20

Therefore:

F only 35 - 20 15

Step 2: Find the number of students who play cricket

Using the formula for the union of two sets:

F ∪ C F C - F ∩ C

We can find the total number of students who play cricket (C) by rearranging the formula:

C F ∪ C F ∩ C - F

Given:

F ∪ C 75 F ∩ C 20 F 35

Substituting the known values:

75 35 C - 20

This simplifies to:

75 15 C

Solving for C:

C 75 - 15 60

Step 3: Calculate the number of students who play only cricket

To find the number of students who play only cricket, we can use the formula:

C only C - F ∩ C

Given:

C 60 F ∩ C 20

Therefore:

C only 60 - 20 40

Summary of Results

Based on the calculations:

Students who play only football 15 Students who play only cricket 40

Final Answer

The final answer is:

Only football: 15 Only cricket: 40

Alternative Method: Step-by-Step Alternative Matrix

An alternative approach to verify the results:

Total students: 75 Students playing both: 20 Students playing only football: 35 - 20 15 Students playing only cricket: 75 - 20 - 15 40

Conclusion

The application of set theory helps us determine how many students play only football, only cricket, or both based on the given data. This method can be extended to similar problems where the union and intersection of sets are involved.