Final answer:
De Morgan's laws in discrete mathematics are two rules that provide equivalences for the negation of logical ANDs and ORs, or set unions and intersections, by negating the operands and switching the operators.
Step-by-step explanation:
De Morgan's laws are two fundamental rules in the field of discrete mathematics, especially in relation to sets and Boolean algebra. These laws are utilized for simplifying and transforming logic expressions and set relationships.
De Morgan's Laws Explained
There are two laws that make up De Morgan's Laws:
- The complement of the union of two sets is equal to the intersection of their complements. Mathematically, this is represented as ¬(A ∪ B) = (¬A) ∩ (¬B).
- The complement of the intersection of two sets is equal to the union of their complements. This can be written as ¬(A ∩ B) = (¬A) ∪ (¬B).
In terms of logic, these laws help in the process of transforming logical ANDs into ORs and vice versa, by flipping the operands and changing the operator when negating an expression. If a logic expression is negated and contains an AND, according to De Morgan's law, it is equivalent to negating each operand separately and joining with an OR, and the same applies but in reverse (replacing OR with AND) when negating an expression containing an OR.