Prove If A → C and B → C and ABC → D, then A → D 数据库

To prove that A → D, we need to show that if A is true, then D must also be true.

Given that ABC → D, we know that if A and B and C are all true, then D must also be true.

We are also given that A → C and B → C, which means that if A is true, then C must also be true, and if B is true, then C must also be true.

Combining these two statements, if A is true, then B is also true (because if A were false, then we wouldn't have enough information to determine whether B is true or false). Therefore, if A and B and C are all true, then D must also be true (because of the ABC → D statement).

Therefore, we can conclude that A → D.

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.

Let A, B,C and D be sets. Prove that (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).

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。