De Morgan’s laws
Move a negation through AND or OR while switching the connective.
De Morgan’s laws is one of 1 logic formula in the discrete math section of this library, and it is used at university level.
Why de morgan’s laws works
For a statement joined by and to be true, both halves must hold, so the only way to break it is for at least one half to fail. At least one is the meaning of or, which is why the connective switches. The second law is the mirror image: an or statement dies only when every option dies, so the negation becomes an and.
What each symbol means
$P,Q$ are propositions.
De Morgan’s laws: when it holds
Apply the negation to every component, including quantified statements.
When it stops applying
It applies only to the outermost connective at each step. In not(P and (Q or R)) the first move gives not P or not(Q or R), and the leftover negation must be pushed in again; stopping halfway leaves a statement that is not fully simplified and often gets misread.
De Morgan’s laws: a worked example
“Not both A and B” means “not A or not B.”
The mistake to avoid
What people do: Students push the negation inside and leave the connective alone, writing not(P and Q) as not P and not Q.
Why it goes wrong: That claims both parts fail, which is much stronger than the original. Take P true and Q false: the real negation is true because the and statement failed, but the rewritten version is false.
Do this instead: Change the connective every time the negation moves in. Not both means at least one is missing, so not(P and Q) becomes not P or not Q, and a two-line truth table settles any doubt.
De Morgan’s laws: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $P,Q$ are propositions.
- Check the conditions before substituting. Apply the negation to every component, including quantified statements.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most discrete math slips.
Where this formula fits
- Subject
- Discrete Math formulas — 11 entries in this library
- Topic
- Logic
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where de morgan’s laws comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Discrete Math Calculator — check your substitution and the value it produces.
- Study discrete math — the subject guide that explains the ideas these formulas compress.
- Discrete Math Practice — questions that make you retrieve the formula instead of recognising it.
- All 11 discrete math formulas — the full grouped reference, or the complete formula library.
Questions about de morgan’s laws
What is the negation of every cat is grey?
Some cat is not grey. Negating a for-all statement turns it into a there-exists statement about the opposite property, which is the quantifier version of these same laws.
Does it work with three or more statements?
Yes, and it applies all the way along. Not(P and Q and R) becomes not P or not Q or not R, because a single failure anywhere is enough to sink the whole chain.
How does the set version look?
The complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements. It is the same statement with or replaced by union and and replaced by intersection.
What is the negation of 3 < x < 5?
That double inequality means x > 3 and x < 5, so its negation is x ≤ 3 or x ≥ 5. Students often write 3 ≥ x ≥ 5 by mistake, which describes no number at all.