Question

In a class of 50 students, 25 watch channel A, 35 watch channel B. Every student watches at least one of these channels. Draw a Venn diagram of this information and hence determine how many students watch. A. both programs. B. channel B only

Answer 1

Sure, let's analyze the problem step-by-step.

1. Let \( A \) be the set of students who watch channel A.
2. Let \( B \) be the set of students who watch channel B.

Given:
- Total number of students, \( |A \cup B| = 50 \)
- Students who watch channel A, \( |A| = 25 \)
- Students who watch channel B, \( |B| = 35 \)

According to the principle of inclusion-exclusion in set theory:

\[ |A \cup B| = |A| + |B| - |A \cap B| \]

Substituting the given values,

\[ 50 = 25 + 35 - |A \cap B| \]
\[ 50 = 60 - |A \cap B| \]
\[ |A \cap B| = 10 \]

A. The number of students who watch both channels (i.e., \( |A \cap B| \)) is 10.

B. To find the number of students who watch only channel B:

\[ \text{Students who watch only channel B} = |B| - |A \cap B| \]
\[ \text{Students who watch only channel B} = 35 - 10 = 25 \]

So, the number of students who watch only channel B is 25.

### Venn Diagram

The Venn diagram would look like this:

- A circle representing set \( A \) (students who watch channel A).
- A circle representing set \( B \) (students who watch channel B).
- The intersection of the two circles represents the 10 students who watch both channels.

Here's how the numbers break down visually:

```
+-------------+
| 15 | 10 | 25 |
| Channel A | | Channel B |
+-------------+
```

- 15 students watch only channel A.
- 10 students watch both channels A and B.
- 25 students watch only channel B.

This verifies the Total:
\[ 15 + 10 + 25 = 50 \]

Therefore, the Venn diagram and the calculations align perfectly with the problem conditions.