How does commutator relate to De Morgan's law in boolean algebra?

In boolean algebra, De Morgan's law states that the negation of a conjunction (AND) of two propositions is equal to the disjunction (OR) of their negations, and vice versa. The two De Morgan's laws can be written as:

~(A & B) = ~A v ~B

~(A v B) = ~A & ~B

The term "commutator" is often used to describe operations that commute, meaning that their order does not affect the result. In boolean algebra, the De Morgan's laws show that negation and conjunction (or disjunction) commute, in the sense that negating the result of a conjunction (or disjunction) is equivalent to first negating the individual terms and then computing the conjunction (or disjunction).

In this sense, the De Morgan's laws are related to the idea of commutativity, as they show that negation and conjunction (or disjunction) can be performed in either order and still produce the same result. The De Morgan's laws are important laws of boolean algebra and are widely used in computer science, logic, and mathematics.

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