Question: 4. In the Venn diagram:
ℰ = {houses on Abbey Road}
C = {houses with a rooftop chimney}
D = {houses with a driveway}
G = {houses with a garden}
a) How many houses have a garden?
b) How many houses have a driveway and a chimney?
c) How many houses have a driveway and a chimney and a garden?
d) How many houses have a garden, but not a driveway or a chimney?
e) How many houses have a driveway and a garden, but not a chimney?
f) How many houses are there on Abbey Road?
1. **State the problem:** We have a Venn diagram with sets:
- $\mathcal{E}$: houses on Abbey Road
- $C$: houses with a rooftop chimney
- $D$: houses with a driveway
- $G$: houses with a garden
Given counts:
- $D$ only = 35
- $C$ only = 20
- $G$ only = 40
- $D \cap C$ only = 6
- $D \cap G$ only = 11
- $C \cap G$ only = 3
- $D \cap C \cap G$ = 5
- Outside all circles but inside $\mathcal{E}$ = 4
2. **Find each requested quantity:**
**a) Houses with a garden ($G$):**
This includes all regions that have $G$:
- $G$ only = 40
- $D \cap G$ only = 11
- $C \cap G$ only = 3
- $D \cap C \cap G$ = 5
Sum:
$$40 + 11 + 3 + 5 = 59$$
**b) Houses with a driveway and a chimney ($D \cap C$):**
This includes the intersection regions of $D$ and $C$:
- $D \cap C$ only = 6
- $D \cap C \cap G$ = 5
Sum:
$$6 + 5 = 11$$
**c) Houses with a driveway, chimney, and garden ($D \cap C \cap G$):**
Given directly:
$$5$$
**d) Houses with a garden but not a driveway or chimney:**
This is $G$ only region:
$$40$$
**e) Houses with a driveway and garden but not a chimney:**
This is $D \cap G$ only region:
$$11$$
**f) Total houses on Abbey Road ($\mathcal{E}$):**
Sum all regions inside $\mathcal{E}$:
$$35 + 20 + 40 + 6 + 11 + 3 + 5 + 4 = 124$$
3. **Summary of answers:**
- a) $59$
- b) $11$
- c) $5$
- d) $40$
- e) $11$
- f) $124$