De Morgan’s Laws

De Morgan’s Laws: How to simplify negated logic statements

De Morgan’s Laws help you rewrite a negation that applies to a group of logical statements. The main idea is that when you negate an AND/OR expression, the operator changes and each part is negated.

1) Remember the two rules

• (\neg (P \land Q) = \neg P \lor \neg Q)
• (\neg (P \lor Q) = \neg P \land \neg Q)

If there are more than two parts, apply the same pattern to every statement inside the brackets.

2) Move the negation inward

Do not leave the negation outside the parentheses if you can simplify it. Replace the outer negation by changing:

• AND to OR, or
• OR to AND, and then negate each statement separately.

3) Simplify the final statement

After rewriting, remove any double negations and write the result in the simplest logical form.

4) Check your answer

A quick check is to test a few cases where the original statement is true or false. The rewritten statement should match the original one in every case.

Example pattern

• Start: (\neg(A \land B))
• Apply the law: (\neg A \lor \neg B)

Keep track of parentheses carefully, especially when the negation applies to a longer expression.