用 De Morgan's Law 化簡下式

$$(\neg A \lor B) \land (\neg A \lor \neg B)$$

首先，我们可以使用 De Morgan's Law 將 $\neg A \lor B$ 和 $\neg A \lor \neg B$ 兩個子句轉換成其否定式：

$$(\neg(\neg A) \land \neg B) \land (\neg(\neg A) \land B)$$

接著，我們可以使用分配律將這兩個子句展開：

$$((A \lor \neg B) \land (A \lor B))$$

因此，原式可以簡化為 $((A \lor \neg B) \land (A \lor B))$。

相关问题

De Morgan s Law

De Morgan's laws are two rules in Boolean algebra that relate to the logical operators "AND" and "OR". The laws are as follows:

1. The complement of the conjunction of two statements is equivalent to the disjunction of their complements. In other words, ~(A ∧ B) ≡ ~A ∨ ~B.

2. The complement of the disjunction of two statements is equivalent to the conjunction of their complements. In other words, ~(A ∨ B) ≡ ~A ∧ ~B.

These laws can be used to simplify logical expressions and to convert between different forms of expressions. For example, if we have the expression ~(A ∧ B), we can use De Morgan's first law to rewrite it as ~A ∨ ~B. Similarly, if we have the expression ~(A ∨ B), we can use De Morgan's second law to rewrite it as ~A ∧ ~B.

De Morgan's Law

De Morgan's Law is a fundamental principle in mathematics and logic that states that the negation of a conjunction (AND) or a disjunction (OR) is equivalent to the disjunction or conjunction, respectively, of the negations of the individual statements. In other words, it provides a way to simplify complex logical expressions by transforming them into equivalent expressions that are easier to work with.

Formally, De Morgan's Law can be stated as follows:

- The negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations of the individual statements. That is, ¬(P ∧ Q) ≡ (¬P) ∨ (¬Q).
- The negation of a disjunction (OR) is equivalent to the conjunction (AND) of the negations of the individual statements. That is, ¬(P ∨ Q) ≡ (¬P) ∧ (¬Q).

For example, suppose we have the statement "It is not raining and it is not cloudy." Using De Morgan's Law, we can transform this into the equivalent statement "It is either raining or cloudy." This is because ¬(not raining AND not cloudy) is the same as saying either it is raining (not not raining) OR it is cloudy (not not cloudy).

De Morgan's Law is a useful tool in many areas of mathematics and computer science, including Boolean algebra, logic circuits, and set theory.