Question 1185451: Consider the following statements
X: All policemen wear uniform;
Y: Civil servants do not wear uniform.
If P={policemen}, T={people who wear uniform} and C={civil servants},
(I). draw a Venn diagram to illustrate X and Y
(II). use the diagram to determine which of the following implications are valid or not valid conclusions from X and Y;
(a) Adu wears uniform ____ Adu is a policeman
(b). Ofei is a policeman_____ Ofei is not a Civil Servants.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to analyze the given statements and draw the Venn diagram:
**(I) Venn Diagram:**
The statements can be represented using a Venn diagram with three overlapping sets: P (Policemen), T (People who wear uniform), and C (Civil Servants).
* **Statement X: All policemen wear a uniform.** This means the entire set P is contained within set T.
* **Statement Y: Civil servants do not wear a uniform.** This means sets C and T have no intersection. They are completely separate.
Here's a visual representation:
```
T (Uniform)
-----------------
| |
| P |
| (Policemen) |
| ------ |
| | | |
| | | |
| | | |
| ------ |
| |
| C |
| (Civil Servants)|
|-----------------|
```
**(II) Valid or Not Valid Conclusions:**
**(a) Adu wears uniform ____ Adu is a policeman:**
* **Not Valid.** Adu wearing a uniform (Adu ∈ T) does *not* necessarily mean Adu is a policeman. The Venn diagram shows that the set T (uniform wearers) is larger than the set P (policemen). Adu could be in the part of T that is *outside* of P. There are many people who wear uniforms who are not policemen.
**(b) Ofei is a policeman _____ Ofei is not a civil servant:**
* **Valid.** Since all policemen wear uniforms (X) and no civil servants wear uniforms (Y), it logically follows that no policeman can be a civil servant. If Ofei is a policeman (Ofei ∈ P), then Ofei must also be in T (Ofei ∈ T). Since C and T do not intersect, Ofei cannot be in C (Ofei ∉ C).
**In summary:**
(a) Adu wears uniform **does not imply** Adu is a policeman.
(b) Ofei is a policeman **implies** Ofei is not a civil servant.
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