Question

the diagram represents three statements about teachers: \\(p\\), \\(q\\), and \\(r\\).

for how many teachers are both \\(p \wedge r\\) true and \\(q\\) false?

2
4
5
9

Explanation:

Translate logical conditions to sets

Using the Logical Connectives knowledge point, we translate the logical statements into set operations.
The condition that \(p \wedge r\) is true means an element must belong to both set \(p\) and set \(r\).
This corresponds to the intersection \(p \cap r\).
The condition that \(q\) is false means the element must not belong to set \(q\).
This corresponds to the complement \(q'\).

Identify the target region

We seek the region representing \(p \cap r \cap q'\).
This is the region inside both circles \(p\) and \(r\), but strictly outside circle \(q\).

Read values from Venn diagram

Looking at the Venn diagram:

• The intersection of circle \(p\) and circle \(r\) contains the regions with numbers \(9\) and \(2\).
• The region with number \(2\) lies inside circle \(q\).
• The region with number \(9\) lies outside circle \(q\).

Calculate the final count

The number of teachers in the region \(p \cap r \cap q'\) is \(9\).

Answer:

• (A) 2
• (B) 4
• (C) 5
• (D) 9 (Correct answer)