In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.
Dalam logika proposional dan aljabar Boolean, Hukum De Morgan, aturan De Morgan, atau teorema De Morgan adalah sepasang aturan penarikan kesimpulan yang menghubungkan konjungsi dan disjungsi melalui negasi.
They are named after Augustus De Morgan, a 19th-century British mathematician.
Aturan ini dinamai dari Augustus De Morgan, seorang matematikawan abad ke-19 berkebangsaan Inggris.
De Morgan's laws are an example of a more general concept of mathematical duality.
Hukum De Morgan adalah contoh dari suatu konsep umum dari dualitas matematika.
\neg (P\land Q)&\vdash (\neg P\lor \neg Q),{\text{and}}\\(\neg P\lor \neg Q)&\vdash \neg (P\land Q).\end{aligned}}} The negation of disjunction rule may be written as: ¬ ( P ∨ Q ) ⊢ ( ¬ P ∧ ¬ Q ) , and ( ¬ P ∧ ¬ Q ) ⊢ ¬ ( P ∨ Q ) .
Q\right){\text{, dan}}\\\neg \!\left(P\land Q\right)&\iff \left(\neg P\right)\lor \left(\neg Q\right)\end{aligned}}} dengan P dan Q merupakan proposisi ¬ menyatakan operator logika negasi (tidak) ∧ menyatakan operator logika konjungsi (dan) ∨ menyatakan operator logika disjungsi (atau) ⟺ adalah simbol metalogika yang berarti "dalam bukti logis, dapat diganti dengan" dan sering kali dibaca sebagai "jika dan hanya jika".
\neg (P\land Q)&\leftrightarrow (\neg P\lor \neg Q),\\\neg (P\lor Q)&\leftrightarrow (\neg P\land \neg Q).\\\end{aligned}}} where P and Q are propositions expressed in some formal system.
\left(\neg Q\right){\text{, dan}}\\\neg \!\left(P\land Q\right)&\leftrightarrow \left(\neg P\right)\lor \left(\neg Q\right)\end{aligned}}} dengan P dan Q adalah proposisi yang diekspresikan dalam suatu sistem formal.
{\overline {A_{1}\land A_{2}\land \ldots \land A_{n}}}={\overline {A_{1}}}\lor {\overline {A_{2}}}\lor \ldots \lor {\overline {A_{n}}},\\{\overline {A_{1}\lor A_{2}\lor \ldots \lor A_{n}}}={\overline {A_{1}}}\land {\overline {A_{2}}}\land \ldots \land {\overline {A_{n}}}.\end{aligned}}} In electrical and computer engineering, De Morgan's laws are commonly written as: ( A ⋅ B ) ¯ ≡ ( A ¯ + B ¯ ) }\equiv ({\overline {A}}+{\overline {B}})} and ( A + B ) ¯ ≡ ( A ¯ ⋅ B ¯ ) , }\equiv ({\overline {A}}\cdot {\overline {B}}),} where: ⋅ is the logical AND, + is the logical OR, the overbar is the logical NOT of what is underneath the overbar.
A + B ¯A ¯ ⋅ B ¯ , dan A ⋅ B ¯A ¯ + B ¯ {\overline {A+B}}&={\overline {A}}\cdot {\overline {B}}{\text{, dan}}\\{\overline {A\cdot B}}&={\overline {A}}+{\overline {B}}\end{aligned}}} dengan A ¯ }} menyatakan operator logika negasi (TIDAK) dari A ⋅ menyatakan operator logika konjungsi (DAN) + menyatakan operator logika disjungsi (ATAU)Pencarian teks
De Morgan's laws hold that these two searches will return the same set of documents:
Hukum De Morgan menyatakan bahwa kedua pencarian ini akan menghasilkan dokumen-dokumen yang sama :
The proof that A ∩ B ¯A ¯ ∪ B ¯ }={\overline {A}}\cup {\overline {B}}} is completed in 2 steps by proving both A ∩ B ¯ ⊆ A ¯ ∪ B ¯ }\subseteq {\overline {A}}\cup {\overline {B}}} and A ¯ ∪ B ¯ ⊆ A ∩ B ¯ }\cup {\overline {B}}\subseteq {\overline {A\cap B}}} .
Untuk membuktikan bahwa A ∪ B ¯A ¯ ∩ B ¯ }={\overline {A}}\cap {\overline {B}}} , maka Akan dibuktikan bahwa A ∪ B ¯ ⊆ A ¯ ∩ B ¯ }\subseteq {\overline {A}}\cap {\overline {B}}} .
If x ∉ A , then x ∈ A ¯ }} , so x ∈ A ¯ ∪ B ¯ }\cup {\overline {B}}} .
Diambil sembarang elemen x ∈ A ∪ B ¯ }} .
Similarly, if x ∉ B , then x ∈ B ¯ }} , so x ∈ A ¯ ∪ B ¯ }\cup {\overline {B}}} .
Diambil sembarang elemen x ∈ A ∪ B ¯ }} .
MathWorld. de Morgan's laws at PlanetMath.
(Inggris) de Morgan's laws di PlanetMath.
Duality in Logic and Language, Internet Encyclopedia of Philosophy.
Dualitas dalam Logika dan Bahasa, Internet Encyclopedia of Philosophy.