Let A, B,C and D be sets. Prove that (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).
时间: 2023-05-19 20:06:35 浏览: 73
答案:我们可以证明两个集合相等的方法是证明它们互相包含。首先,假设 (x, y) 属于 (A × B) ∩ (C × D),那么 x 属于 A 且 y 属于 B,同时 x 属于 C 且 y 属于 D。因此,x 属于 A ∩ C 且 y 属于 B ∩ D,即 (x, y) 属于 (A ∩ C) × (B ∩ D)。因此,(A × B) ∩ (C × D) 包含于 (A ∩ C) × (B ∩ D)。
反过来,假设 (x, y) 属于 (A ∩ C) × (B ∩ D),那么 x 属于 A 且 x 属于 C,同时 y 属于 B 且 y 属于 D。因此,(x, y) 属于 A × B 且 (x, y) 属于 C × D,即 (x, y) 属于 (A × B) ∩ (C × D)。因此,(A ∩ C) × (B ∩ D) 包含于 (A × B) ∩ (C × D)。
因此,我们证明了两个集合互相包含,从而证明了它们相等。
相关问题
Prove If A → C and B → C and ABC → D, then A → D
To prove that A → D, we need to show that if A is true, then D must also be true.
We know that ABC → D, which means that if A, B, and C are all true, then D must also be true. Since A → C and B → C, we know that if A and B are true, then C must also be true.
Therefore, if A and B are true, then C is true, and if C is true, then D is true. So, if A is true, then D must also be true, and we have proved that A → D.
解答下题Let A = {a, b, c} and B = P(A). 1) Prove that [B; ∪ , ∩ ,~] is a Boolean algebra. 2) Write out the operation tables for the Boolean algebra.
1) To prove that [B; ∪ , ∩ ,~] is a Boolean algebra, we need to show that it satisfies the following axioms:
- Closure under union and intersection: For any two sets X, Y ∈ B, X ∪ Y and X ∩ Y ∈ B.
- Associativity of union and intersection: For any three sets X, Y, Z ∈ B, (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z) and (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z).
- Commutativity of union and intersection: For any two sets X, Y ∈ B, X ∪ Y = Y ∪ X and X ∩ Y = Y ∩ X.
- Distributivity of intersection over union and vice versa: For any three sets X, Y, Z ∈ B, X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z) and X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z).
- Identity elements of union and intersection: There exist two sets 0, 1 ∈ B such that for any set X ∈ B, X ∪ 0 = X and X ∩ 1 = X.
- Complementation: For any set X ∈ B, there exists a set ¬X ∈ B such that X ∪ ¬X = 1 and X ∩ ¬X = 0.
All of these axioms can be easily verified for [B; ∪ , ∩ ,~], therefore it is a Boolean algebra.
2) The operation tables for the Boolean algebra [B; ∪ , ∩ ,~] are:
| ∪ | a b c ∅
--|----------------
a | a a a a
b | a b b b
c | a b c c
∅ | a b c ∅
| ∩ | a b c ∅
--|----------------
a | a b c ∅
b | b b ∅ ∅
c | c ∅ c ∅
∅ | ∅ ∅ ∅ ∅
| ~ | a b c ∅
--|------------------------
| ∅ {b,c} {a,c} {a,b}
a | {b,c} ∅ {b} {c}
b | {a,c} {a} ∅ {c}
c | {a,b} {a} {b} ∅
∅ | {a,b,c} {a,b,c} {a,b,c} ∅
Note: ∅ represents the empty set, and {} represents the set with no elements.