View source for De Morgan's laws

[[File:Demorganlaws.svg|thumb|De Morgan's laws represented with [[Venn diagrams]]. In each case, the resultant set is the set of all points in any shade of blue.]]

{{Transformation rules}}

In [[propositional calculus|propositional logic]] and [[boolean algebra]], '''De Morgan's laws'''<ref>Copi and Cohen{{Full citation needed|date=January 2015}}</ref><ref>{{citation|first=Patrick J.|last=Hurley|title=A Concise Introduction to Logic|edition=12th|publisher=Cengage Learning|year=2015|isbn=978-1-285-19654-1}}</ref><ref>Moore and Parker{{Full citation needed|date=January 2015}}</ref> are a pair of transformation rules that are both [[Validity (logic)|valid]] [[rule of inference|rules of inference]]. They are named after [[Augustus De Morgan]], a 19th-century British mathematician. The rules allow the expression of [[Logical conjunction|conjunctions]] and [[Logical disjunction|disjunctions]] purely in terms of each other via [[logical negation|negation]].

The rules can be expressed in English as:
:the negation of a disjunction is the conjunction of the negations; and
:the negation of a conjunction is the disjunction of the negations;
or
:the [[Complement (set theory)|complement]] of the union of two sets is the same as the intersection of their complements; and
:the complement of the intersection of two sets is the same as the union of their complements.
or
:not (A or B) = not A and not B; and
:not (A and B) = not A or not B

In [[set theory]] and [[Boolean algebra (logic)|Boolean algebra]], these are written formally as
:<math>\begin{align}
  \overline{A \cup B} &= \overline{A} \cap \overline{B}, \\
  \overline{A \cap B} &= \overline{A} \cup \overline{B},
\end{align}</math>
where
* ''A'' and ''B'' are sets,
* {{overline|''A''}} is the complement of A,
* ∩ is the [[Intersection (set theory)|intersection]], and
* ∪ is the [[Union (set theory)|union]].

In [[formal language]], the rules are written as
:<math>\neg(P\lor Q)\iff(\neg P)\land(\neg Q),</math>
and
:<math>\neg(P\land Q)\iff(\neg P)\lor(\neg Q)</math>
where
* ''P'' and ''Q are propositions,''
* [[Negation|<math>\neg</math>]] is the negation logic operator (NOT),
* [[Logical_conjunction|<math>\land</math>]] is the conjunction logic operator (AND),
* [[Logical_disjunction|<math>\lor</math>]] is the disjunction logic operator (OR),
* [[If_and_only_if|<math>\iff</math>]] is a [[metalogic]]al symbol meaning "can be replaced in a [[formal proof|logical proof]] with".

Applications of the rules include simplification of logical [[Expression (computer science)|expressions]] in [[computer program]]s and digital circuit designs. De Morgan's laws are an example of a more general concept of [[duality (mathematics)|mathematical duality]].

==Formal notation==
The ''negation of conjunction'' rule may be written in [[sequent]] notation:
:<math>\neg(P \land Q) \vdash (\neg P \lor \neg Q).</math>

The ''negation of disjunction'' rule may be written as:
:<math>\neg(P \lor Q) \vdash (\neg P \land \neg Q).</math>

In [[Rule of inference|rule form]]: ''negation of conjunction''
:<math>\frac{\neg (P \land Q)}{\therefore \neg P \lor \neg Q}</math>

and ''negation of disjunction''
:<math>\frac{\neg (P \lor Q)}{\therefore \neg P \land \neg Q}</math>

and expressed as a truth-functional [[Tautology (logic)|tautology]] or [[theorem]] of propositional logic:

:<math>\begin{align}
  \neg (P \land Q) &\to (\neg P \lor \neg Q), \\
   \neg (P \lor Q) &\to (\neg P \land \neg Q),
\end{align}</math>

where <math>P</math> and <math>Q</math> are propositions expressed in some formal system.

===Substitution form===
De Morgan's laws are normally shown in the compact form above, with negation of the output on the left and negation of the inputs on the right. A clearer form for substitution can be stated as:

:<math>\begin{align}
  (P \land Q) &\equiv \neg (\neg P \lor \neg Q), \\
   (P \lor Q) &\equiv \neg (\neg P \land \neg Q).
\end{align}</math>

This emphasizes the need to invert both the inputs and the output, as well as change the operator, when doing a substitution.

===Set theory and Boolean algebra===
In set theory and [[Boolean algebra (logic)|Boolean algebra]], it is often stated as "union and intersection interchange under complementation",<ref>''Boolean Algebra'' by R. L. Goodstein. {{ISBN|0-486-45894-6}}</ref> which can be formally expressed as:
:<math>\begin{align}
  \overline{A \cup B} &= \overline{A} \cap \overline{B}, \\
  \overline{A \cap B} &= \overline{A} \cup \overline{B},
\end{align}</math>

where:
* {{overline|''A''}} is the negation of A, the [[overline]] being written above the terms to be negated,
* ∩ is the [[Intersection (set theory)|intersection]] operator (AND),
* ∪ is the [[Union (set theory)|union]] operator (OR).

The generalized form is:
: <math>\begin{align}
  \overline{\bigcap_{i \in I} A_{i}} &\equiv \bigcup_{i \in I} \overline{A_{i}}, \\
  \overline{\bigcup_{i \in I} A_{i}} &\equiv \bigcap_{i \in I} \overline{A_{i}},
\end{align}</math>

where {{math|''I''}} is some, possibly uncountable, indexing set.

In set notation, De Morgan's laws can be remembered using the [[mnemonic]] "break the line, change the sign".<ref>[https://books.google.com/books?id=NdAjEDP5mDsC&pg=PA81&lpg=PA81&dq=break+the+line+change+the+sign&source=web&ots=BtUl4oQOja&sig=H1Wz9e6Uv_bNeSbTvN6lr3s47PQ#PPA81,M1 ''2000 Solved Problems in Digital Electronics''] by S. P. Bali</ref>

===Engineering===
In [[electrical and computer engineering]], De Morgan's laws are commonly written as:
: <math>\overline{A \cdot B} \equiv \overline {A} + \overline {B}</math>

and
: <math>\overline{A + B} \equiv \overline {A} \cdot \overline {B},</math>

where:
* <math> \cdot </math> is the logical AND,
* <math>+</math> is the logical OR,
* the {{overline|overbar}} is the logical NOT of what is underneath the overbar.

===Text searching===
De Morgan’s laws commonly apply to text searching using Boolean operators AND, OR, and NOT.  Consider a set of documents containing the words “cars” and “trucks”. De Morgan’s laws hold that these two searches will return the same set of documents:

:Search A: NOT (cars OR trucks)
:Search B: (NOT cars) AND (NOT trucks)

The corpus of documents containing “cars” or “trucks” can be represented by four documents:
:Document 1: Contains only the word “cars”.
:Document 2: Contains only “trucks”.
:Document 3: Contains both “cars” and “trucks”.
:Document 4: Contains neither “cars” nor “trucks”.

To evaluate Search A, clearly the search “(cars OR trucks)” will hit on Documents 1, 2, and 3. So the negation of that search (which is Search A) will hit everything else, which is Document 4.

Evaluating Search B, the search “(NOT cars)” will hit on documents that do not contain “cars”, which is Documents 2 and 4.  Similarly the search “(NOT trucks)” will hit on Documents 1 and 4. Applying the AND operator to these two searches (which is Search B) will hit on the documents that are common to these two searches, which is Document 4.

A similar evaluation can be applied to show that the following two searches will return the same set of documents (Documents 1, 2, 4):
:Search C: NOT (cars AND trucks),
:Search D: (NOT cars) OR (NOT trucks).

==History==
The laws are named after [[Augustus De Morgan]] (1806–1871),<ref>''[http://www.mtsu.edu/~phys2020/Lectures/L19-L25/L3/DeMorgan/body_demorgan.html DeMorgan’s Theorems]'' at mtsu.edu</ref> who introduced a formal version of the laws to classical [[propositional logic]]. De Morgan's formulation was influenced by algebraization of logic undertaken by [[George Boole]], which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by [[Aristotle]], and was known to Greek and Medieval logicians.<ref>Bocheński's ''History of Formal Logic''</ref> For example, in the 14th century, [[William of Ockham]] wrote down the words that would result by reading the laws out.<ref>William of Ockham, ''Summa Logicae'', part II, sections 32 and 33.</ref> [[Jean Buridan]], in his ''Summulae de Dialectica'', also describes rules of conversion that follow the lines of De Morgan's laws.<ref>Jean Buridan, ''Summula de Dialectica''. Trans. Gyula Klima. New Haven: Yale University Press, 2001. See especially Treatise 1, Chapter 7, Section 5. {{ISBN|0-300-08425-0}}</ref> Still, De Morgan is given credit for stating the laws in the terms of modern formal logic, and incorporating them into the language of logic. De Morgan's laws can be proved easily, and may even seem trivial.<ref>[http://www.engr.iupui.edu/~orr/webpages/cpt120/mathbios/ademo.htm Augustus De Morgan (1806–1871)] {{webarchive|url=https://web.archive.org/web/20100715185655/http://www.engr.iupui.edu/~orr/webpages/cpt120/mathbios/ademo.htm |date=2010-07-15 }} by Robert H. Orr</ref> Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.

==Informal proof==
De Morgan's theorem may be applied to the negation of a [[disjunction]] or the negation of a [[Logical conjunction|conjunction]] in all or part of a formula.

===Negation of a disjunction===
In the case of its application to a disjunction, consider the following claim: "it is false that either of A or B is true", which is written as:
:<math>\neg(A\lor B).</math>
In that it has been established that ''neither'' A nor B is true, then it must follow that both A is not true [[logical AND|and]] B is not true, which may be written directly as:
:<math>(\neg A)\wedge(\neg B).</math>
If either A or B ''were'' true, then the disjunction of A and B would be true, making its negation false. Presented in English, this follows the logic that "since two things are both false, it is also false that either of them is true".

Working in the opposite direction, the second expression asserts that A is false and B is false (or equivalently that "not A" and "not B" are true). Knowing this, a disjunction of A and B must be false also. The negation of said disjunction must thus be true, and the result is identical to the first claim.

===Negation of a conjunction===
The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale.  Consider the following claim: "it is false that A and B are both true", which is written as:
:<math>\neg(A\land B).</math>
In order for this claim to be true, either or both of A or B must be false, for if they both were true, then the conjunction of A and B would be true, making its negation false. Thus, [[inclusive or|one (at least) or more]] of A and B must be false (or equivalently, one or more of "not A" and "not B" must be true). This may be written directly as,
:<math>(\neg A)\lor(\neg B).</math>
Presented in English, this follows the logic that "since it is false that two things are both true, at least one of them must be false".

Working in the opposite direction again, the second expression asserts that at least one of "not A" and "not B" must be true, or equivalently that at least one of A and B must be false. Since at least one of them must be false, then their conjunction would likewise be false. Negating said conjunction thus results in a true expression, and this expression is identical to the first claim.

==Formal proof==
Here we use <math>A^\complement</math>to denote the complement of A. The proof that <math>(A\cap B)^\complement = A^\complement \cup B^\complement</math> is completed in 2 steps by proving both <math>(A\cap B)^\complement \subseteq A^\complement \cup B^\complement</math> and <math>A^\complement \cup B^\complement \subseteq (A\cap B)^\complement</math>. 

===Part 1===

Let <math>x \in (A \cap B)^\complement</math>. Then, <math>x \not\in A \cap B</math>.

Because <math>A \cap B = \{y | y \in A \land y \in B\}</math>, it must be the case that <math>x \not\in A</math> or <math>x \not\in B</math>.

If <math>x \not\in A</math>, then <math>x \in A^\complement</math>, so <math>x \in A^\complement \cup B^\complement</math>.

Similarly, if <math>x \not\in B</math>, then <math>x \in B^\complement</math>, so <math>x \in A^\complement\cup B^\complement</math>.

Thus, <math>\forall x( x \in (A\cap B)^\complement \rarr x \in A^\complement \cup B^\complement)</math>;

that is, <math>(A\cap B)^\complement \subseteq A^\complement \cup B^\complement</math>.

===Part 2===

To prove the reverse direction, let <math>x \in A^\complement \cup B^\complement</math>, and for contradiction assume <math>x \not\in (A\cap B)^\complement</math>.

Under that assumption, it must be the case that <math>x \in A\cap B</math>,

so it follows that <math>x \in A</math> and <math>x \in B</math>, and thus <math>x \not\in A^\complement</math> and <math>x \not\in B^\complement</math>.

However, that means <math>x \not\in A^\complement \cup B^\complement</math>, in contradiction to the hypothesis that <math>x \in A^\complement \cup B^\complement</math>,

therefore, the assumption <math>x \not\in (A\cap B)^\complement</math> must not be the case, meaning that <math>x \in (A\cap B)^\complement</math>.

Hence, <math>\forall x( x \in A^\complement \cup B^\complement \rarr x \in (A\cap B)^\complement)</math>,

that is, <math>A^\complement \cup B^\complement \subseteq (A\cap B)^\complement</math>.

===Conclusion===

If <math>A^\complement \cup B^\complement \subseteq (A\cap B)^\complement</math> ''and'' <math>(A \cap B)^\complement \subseteq A^\complement \cup B^\complement</math>, then <math>(A\cap B)^\complement = A^\complement \cup B^\complement</math>; this concludes the proof of De Morgan's law.

The other De Morgan's law, <math>(A \cup B)^\complement = A^\complement \cap B^\complement</math>, is proven similarly.

==Generalising De Morgan duality==
[[File:DeMorgan Logic Circuit diagram DIN.svg|thumb|De Morgan's Laws represented as a circuit with logic gates]]

In extensions of classical propositional logic, the duality still holds (that is, to any logical operator one can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on [[classical logic]], namely the existence of [[negation normal form]]s: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in [[digital circuit]] design, where it is used to manipulate the types of [[logic gate]]s, and in formal logic, where it is needed to find the [[conjunctive normal form]] and [[disjunctive normal form]] of a formula.  Computer programmers use them to simplify or properly negate complicated [[Conditional (programming)|logical conditions]]. They are also often useful in computations in elementary [[probability theory]].

Let one define the dual of any propositional operator P(''p'', ''q'', ...) depending on elementary propositions ''p'', ''q'', ... to be the operator <math>\mbox{P}^d</math> defined by

:<math>\mbox{P}^d(p, q, ...) = \neg P(\neg p, \neg q, \dots).</math>

==Extension to predicate and modal logic==

This duality can be generalised to quantifiers, so for example the [[universal quantifier]] and [[existential quantifier]] are duals:

:<math> \forall x \, P(x) \equiv \neg [ \exists x \, \neg P(x)] </math>

:<math> \exists x \, P(x) \equiv \neg [ \forall x \, \neg P(x)] </math>

To relate these quantifier dualities to the De Morgan laws, set up a [[model theory|model]] with some small number of elements in its domain ''D'', such as

:''D'' = {''a'', ''b'', ''c''}.

Then

:<math> \forall x \, P(x) \equiv P(a) \land P(b) \land P(c) </math>

and

:<math> \exists x \, P(x) \equiv P(a) \lor P(b) \lor P(c).</math>

But, using De Morgan's laws,

:<math> P(a) \land P(b) \land P(c) \equiv \neg (\neg P(a) \lor \neg P(b) \lor \neg P(c)) </math>

and

:<math> P(a) \lor P(b) \lor P(c) \equiv \neg (\neg P(a) \land \neg P(b) \land \neg P(c)), </math>

verifying the quantifier dualities in the model.

Then, the quantifier dualities can be extended further to [[modal logic]], relating the box ("necessarily") and diamond ("possibly") operators:

:<math> \Box p \equiv \neg \Diamond \neg p, </math>
:<math> \Diamond p \equiv \neg \Box \neg p.</math>

In its application to the [[alethic modalities]] of possibility and necessity, [[Aristotle]] observed this case, and in the case of [[normal modal logic]], the relationship of these modal operators to the quantification can be understood by setting up models using [[Kripke semantics]].

==See also==
* [[Isomorphism]] – NOT operator as isomorphism between positive logic and negative logic
* [[List of Boolean algebra topics]]
* [[Positive logic]]

==References==
{{reflist}}

==External links==
* {{springer|title=Duality principle|id=p/d034130}}
* {{MathWorld | urlname=deMorgansLaws | title=de Morgan's Laws}}
* {{PlanetMath | urlname=DeMorgansLaws | title=de Morgan's laws | id=2308}}

{{Classical logic}}
{{Set theory}}

[[Category:Boolean algebra]]
[[Category:Duality theories]]
[[Category:Rules of inference]]
[[Category:Articles containing proofs]]
[[Category:Theorems in propositional logic]]