De Morgan's law states that the negation of "P and Q" is logically equivalent to:

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Not P or not Q. De Morgan's first law states that the negation of a conjunction is the disjunction of the negations: ¬(P ∧ Q) ≡ (¬P ∨ ¬Q). In plain English: "It is not the case that both P and Q are true" is equivalent to "Either P is false, or Q is false (or both)." For example, the negation of "It is raining AND cold" is "It is not raining OR it is not cold."

De Morgan's second law states the complementary relationship: ¬(P ∨ Q) ≡ (¬P ∧ ¬Q). The negation of a disjunction is the conjunction of the negations: "It is not the case that P or Q" is equivalent to "P is false AND Q is false." Together, these laws show that AND and OR are "dual" operations  -  negating one gives the other with negated components.

"Not P and not Q" (¬P ∧ ¬Q) would be the negation of "P or Q," not "P and Q"  -  this is De Morgan's second law, not the first. "P or not Q" (P ∨ ¬Q) negates only Q, not both. "Not P and Q" (¬P ∧ Q) negates only P. De Morgan's laws are fundamental in logic and are essential for the CLEP exam. They also apply to set operations: (A ∩ B)' = A' ∪ B' and (A ∪ B)' = A' ∩ B'. The key pattern to remember: when you negate, the AND becomes OR (and vice versa), and each component gets negated.

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