Logische Konjunktion

Logische Konjunktion
Venn diagram of Logical conjunction
Logik -Tor AND ANSI.svg
Normale Formen
Zhegalkin Polynom
Posts Gitter
0-Präserving Jawohl
1-Präserving Jawohl
Monoton nein
Befriedigung nein
Venn-Diagramm von

Im Logik, Mathematik und Linguistik, Und () ist der Wahrheitsfunktional operator of logische Konjunktion; das und of a set of operands is true if and only if alle of its operands are true. Das Logische Binde that represents this operator is typically written as oder .[1][2]

is true if and only if ist wahr und ist wahr.

An operand of a conjunction is a Konjunkt.

Beyond logic, the term "conjunction" also refers to similar concepts in other fields:


Und is usually denoted by an infix operator: in mathematics and logic, it is denoted by ,[2] & oder × ; in electronics, ; and in programming languages, &, &&, oder und. Im Jan łukasiewicz's Präfixnotation für die Logik, der Bediener ist K, for Polish koniunkcja.[3]


Logische Konjunktion ist ein Betrieb auf zwei logische Werte, typically the values of two Aussagen, that produces a value of Stimmt dann und nur dann, wenn both of its operands are true.[1][2]

The conjunctive Identität is true, which is to say that AND-ing an expression with true will never change the value of the expression. In keeping with the concept of leere Wahrheit, when conjunction is defined as an operator or function of arbitrary Arity, the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result true.


Conjunctions of the arguments on the left — The Stimmt bisschens form a Sierpinski -Dreieck.

Das Wahrheitstabelle von :[1][2]


Definiert von anderen Betreibern

In systems where logical conjunction is not a primitive, it may be defined as[4]


Introduction and elimination rules

As a rule of inference, Konjunktion Einführung is a classically gültig, einfach Argumentform. The argument form has two premises, A und B. Intuitively, it permits the inference of their conjunction.

Deswegen, A und B.

oder in logical operator Notation:

Here is an example of an argument that fits the form Konjunktion Einführung:

Bob likes apples.
Bob likes oranges.
Therefore, Bob likes apples and Bob likes oranges.

Konjunktion der Eliminierung is another classically gültig, einfach Argumentform. Intuitively, it permits the inference from any conjunction of either element of that conjunction.

A und B.
Deswegen, A.

...oder alternativ,

A und B.
Deswegen, B.

Im logical operator Notation:

...oder alternativ,



A conjunction is proven false by establishing either oder . In terms of the object language, this reads

This formula can be seen as a special case of

Wenn is a false proposition.

Other proof strategies

Wenn impliziert dann beides ebenso gut wie prove the conjunction false:

In other words, a conjunction can actually be proven false just by knowing about the relation of its conjuncts, and not necessary about their truth values.

This formula can be seen as a special case of

Wenn is a false proposition.

Either of the above are constructively valid proofs by contradiction.


Amtativität: Jawohl

Venn0001.svg          Venn0001.svg

Assoziativität: Jawohl

Venn 0101 0101.svg Venn 0000 0011.svg          Venn 0000 0001.svg          Venn 0001 0001.svg Venn 0000 1111.svg

Verbreitung: with various operations, especially with oder

Venn 0101 0101.svg Venn 0011 1111.svg          Venn 0001 0101.svg          Venn 0001 0001.svg Venn 0000 0101.svg

idempotenz: Jawohl

Venn01.svg Venn01.svg          Venn01.svg

monotonicity: Jawohl

Venn 1011 1011.svg          Venn 1111 1011.svg          Venn 0000 0101.svg Venn 0000 0011.svg

truth-preserving: yes
When all inputs are true, the output is true.

Venn0001.svg          Venn0001.svg
(getestet werden)

falsehood-preserving: yes
When all inputs are false, the output is false.

Venn0001.svg          Venn0111.svg
(getestet werden)

Walsh spectrum: (1,-1,-1,1)

NichtLinearität: 1 (the function is gebogen))

If using binär values for true (1) and false (0), then logische Konjunktion works exactly like normal arithmetic Multiplikation.

Applications in computer engineering

In high-level computer programming and Digitale Elektronik, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "UND", an algebraic multiplication, or the ampersand symbol & (sometimes doubled as in &&). Many languages also provide Kurzschluss control structures corresponding to logical conjunction.

Logical conjunction is often used for bitwise operations, where 0 corresponds to false and 1 to true:

  • 0 AND 0 = 0,
  • 0 AND 1 = 0,
  • 1 AND 0 = 0,
  • 1 AND 1 = 1.

The operation can also be applied to two binary Wörter angesehen als Bitstrings of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. Zum Beispiel:

  • 11000110 AND 10100011 = 10000010.

This can be used to select part of a bitstring using a bit mask. Zum Beispiel, 10011101 AND 00001000 = 00001000 extracts the fifth bit of an 8-bit bitstring.

Im Computernetzwerk, bit masks are used to derive the network address of a Subnetz within an existing network from a given IP Adresse, by ANDing the IP address and the Subnetzmaske.

Logical conjunction "UND" is also used in Sql operations to form Datenbank Abfragen.

Das Curry -Howard -Korrespondenz relates logical conjunction to product types.

Set-theoretic correspondence

The membership of an element of an intersection set in Mengenlehre is defined in terms of a logical conjunction: xAB dann und nur dann, wenn (xA) ∧ (xB). Through this correspondence, set-theoretic intersection shares several properties with logical conjunction, such as Assoziativität, Amtativität und idempotenz.

Natürliche Sprache

As with other notions formalized in mathematical logic, the logical conjunction und is related to, but not the same as, the grammatical conjunction und in natural languages.

Englisch "und" hat Eigenschaften, die nicht durch logische Konjunktion erfasst wurden.Zum Beispiel "und" impliziert manchmal Ordnung mit dem Sinn von "dann".Zum Beispiel bedeutet "Sie haben geheiratet und hatten ein Kind" im gemeinsamen Diskurs, dass die Ehe vor dem Kind kam.

The word "and" can also imply a partition of a thing into parts, as "The American flag is red, white, and blue." Here, it is not meant that the flag is auf einmal Rot, Weiß und Blau, sondern dass es einen Teil jeder Farbe hat.

Siehe auch


  1. ^ a b c "Konjunktion, Negation und Disjunktion". Philosophie.Lander.edu. Abgerufen 2020-09-02.
  2. ^ a b c d "2.2: Konjunktionen und Disjunktionen". Mathematiklibretenexte. 2019-08-13. Abgerufen 2020-09-02.
  3. ^ Józef Maria Bocheński (1959), Eine Précis mathematischer Logik, übersetzt von Otto Bird aus den französischen und deutschen Ausgaben, Dordrecht, South Holland: D. Reidel, passim.
  4. ^ Smith, Peter. "Arten von Proof -Systemen" (PDF). p. 4.

Externe Links