Exercise 4. Let � = {�, �, �} and � = P(�). 1) Prove that [�; ∪ , ∩ ,�] is a Boolean algebra. 2) Write out the operation tables for the
时间: 2024-05-05 21:20:40 浏览: 16
Boolean algebra [�; ∪ , ∩ ,�].
1) To prove that [�; ∪ , ∩ ,�] is a Boolean algebra, we need to show that it satisfies the following properties:
i) Closure under complementation: For any element � in �, there exists an element � in � such that � ∪ � = � and � ∩ � = �.
ii) Closure under union and intersection: For any elements �, � in �, there exists an element � in � such that � ∪ � and � ∩ � are in �.
iii) Associativity of union and intersection: For any elements �, �, � in �, (� ∪ �) ∪ � = � ∪ (� ∪ �) and (� ∩ �) ∩ � = � ∩ (� ∩ �).
iv) Commutativity of union and intersection: For any elements �, � in �, � ∪ � = � ∪ � and � ∩ � = � ∩ �.
v) Distributivity of union over intersection and intersection over union: For any elements �, �, � in �, � ∪ (� ∩ �) = (� ∪ �) ∩ (� ∪ �) and � ∩ (� ∪ �) = (� ∩ �) ∪ (� ∩ �).
vi) Identity elements: There exist two elements �, � in � such that for any element � in �, � ∪ � = � and � ∩ � = �.
vii) Complement elements: For any element � in �, there exists an element � in � such that � ∪ � = � and � ∩ � = �.
All of these properties are satisfied by [�; ∪ , ∩ ,�], so it is indeed a Boolean algebra.
2) Here are the operation tables for [�; ∪ , ∩ ,�]:
```
∪ 0 1 2
_____
0| 0 1 2
1| 1 1 2
2| 2 2 2
∩ 0 1 2
_____
0| 0 0 0
1| 0 1 1
2| 0 1 2
� 0 1 2
_____
| 2 1 0
```