Topic: Sets and operations on sets *Question:* Forty traders were asked whether they had shops in Accra (A), Takoradi (T) or Kumasi (K). None of them had shops in all three places. 27 of them had shops in only one place: 7 in A, 15 in T and 5 in K. Of those who had shops in two places, only one did not have a shop in T, and two did not have a shop in K. 10 of the traders had shops in K. How many of the traders did not have a shop in any of the three places?

Answer 1

To solve this question, let's use the principle of inclusion-exclusion.

Let's define the following sets:
- A: Traders with shops in Accra
- T: Traders with shops in Takoradi
- K: Traders with shops in Kumasi

We have the following information:
- |A| = 7 (7 traders have shops in Accra only)
- |T| = 15 (15 traders have shops in Takoradi only)
- |K| = 10 (10 traders have shops in Kumasi)

We also know that:
- |A ∩ T ∩ K| = 0 (none of the traders have shops in all three places)

We can now use the principle of inclusion-exclusion to find the number of traders who do not have a shop in any of the three places:

|A ∪ T ∪ K| = |A| + |T| + |K| - |A ∩ T| - |A ∩ K| - |T ∩ K| + |A ∩ T ∩ K|

Since |A ∩ T| = 1 (only one trader has shops in both Accra and Takoradi), and |T ∩ K| = 2 (two traders have shops in both Takoradi and Kumasi), we can substitute these values into the equation:

|A ∪ T ∪ K| = 7 + 15 + 10 - 1 - |A ∩ K| - 2 + 0

We still need to find |A ∩ K|. We can find this number by subtracting the total number of traders who have shops only in Accra (7) and those who have shops only in Kumasi (5) from the total number of traders who have shops in Kumasi (10):

|A ∩ K| = |K| - (|A| - |A ∩ T ∩ K|) - (|K| - |A ∩ T ∩ K|) = 10 - (7 - 0) - (10 - 0) = 3

Substituting this value back into the equation, we get:

|A ∪ T ∪ K| = 7 + 15 + 10 - 1 - 3 - 2 + 0 = 26

Therefore, 26 traders do not have a shop in any of the three places.