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?
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.
To find the number of traders who did not have a shop in any of the three places (A, T, K), we can use the principle of inclusion-exclusion.
Let's break down the information given step-by-step:
1. Total traders surveyed: 40
2. Traders with shops in only one place:
- 7 traders had shops in Accra (A)
- 15 traders had shops in Takoradi (T)
- 5 traders had shops in Kumasi (K)
Therefore, the total number of traders with shops in only one place is 7 + 15 + 5 = 27.
3. Traders with shops in two places:
- Only one trader did not have a shop in Takoradi (T)
- Two traders did not have a shop in Kumasi (K)
This means that the remaining traders with shops in two places, must have a shop in Takoradi (T) but not in Kumasi (K). So, the total number of traders with shops in two places is 1 + 2 = 3.
4. Traders with shops in Kumasi (K): 10
Now, let's calculate the number of traders who did not have a shop in any of the three places using the principle of inclusion-exclusion.
Total traders without a shop in any of the three places = Total traders - Traders with shops in only one place - Traders with shops in two places + Traders with shops in Kumasi (K)
Total traders without a shop in any of the three places = 40 - 27 - 3 + 10
Total traders without a shop in any of the three places = 20
Therefore, the number of traders who did not have a shop in any of the three places is 20.