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Logical conjunction
please note:
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factoids | |
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Ð”Ð° | ÐÐµÑ‚|Ð½ÐµÑ‚=no}} | Ð”Ð° | ÐÐµÑ‚|Ð½ÐµÑ‚=no}} | Ð”Ð° | ÐÐµÑ‚|Ð½ÐµÑ‚=no}} | Ð”Ð° | ÐÐµÑ‚|Ð½ÐµÑ‚=no}} | Ð”Ð° | ÐÐµÑ‚|Ð½ÐµÑ‚=no}}}}{{distinguish|Caret {{!}} Circumflex Agent (^)|Lambda {{!}} Capital Lambda (Î›)|Turned V {{!}} Turned V (Î›)}}(File:Venn 0000 0001.svg|220px|thumb|Venn diagram of A land B land C)In logic, mathematics and linguistics, And (âˆ§) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is typically written as {{math| âˆ§ }} or {{math| â‹… }}. A land B is true only if A is true and B is true. An operand of a conjunction is a conjunct.The term "logical conjunction" is also used for the greatest lower bound in lattice theory.Related concepts in other fields are:
NotationAnd is usually denoted by an infix operator: in mathematics and logic, it is denoted by {{math| âˆ§ }}, {{math|&}} or {{math| Ã— }}; in electronics, {{math| â‹… }}; and in programming languages, &, &&, or and. In Jan Åukasiewicz's prefix notation for logic, the operator is K, for Polish koniunkcja.JÃ³zef Maria BocheÅ„ski (1959), A PrÃ©cis of Mathematical Logic, translated by Otto Bird from the French and German editions, Dordrecht, North Holland: D. Reidel, passim.DefinitionLogical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.The conjunctive identity is 1, which is to say that AND-ing an expression with 1 will never change the value of the expression. In keeping with the concept of vacuous truth, 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 1.Truth tableFile:Multigrade operator AND.svg|thumb|Conjunctions of the arguments on the left â€” The true bits form a Sierpinski triangleSierpinski triangleThe truth table of A land B:{| class="wikitable" style="text-align:center; background-color: #ddffdd;" |
| A wedge B | |||||||||
| T | |||||||||
| F | |||||||||
| F | |||||||||
| F |
Defined by other operators
In systems where logical conjunction is not a primitive, it may be defined asWEB,weblink Types of proof system, Smith, Peter, 4,
A land B = neg(A to neg B)
or
A land B = neg(neg A lor neg B).
Introduction and elimination rules
As a rule of inference, conjunction Introduction is a classically valid, simple argument form. The argument form has two premises, A and B. Intuitively, it permits the inference of their conjunction.
A,
B.
Therefore, A and B.
or in logical operator notation:
A,
B
vdash A land B
Here is an example of an argument that fits the form conjunction introduction:
Bob likes apples.
Bob likes oranges.
Therefore, Bob likes apples and oranges.
Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction.
A and B.
Therefore, A.
...or alternately,
A and B.
Therefore, B.
In logical operator notation:
A land B
vdash A
...or alternately,
A land B
vdash B
Negation
Definition
A conjunction Aland B is be proven false by establishing either neg A or neg B. In terms of the object language, this reads
neg Atoneg(Aland B)
This formula can be seen as a special case of
(Ato C) to ( (Aland B)to C )
when C is a false proposition.Other proof strategies
If A implies neg B, then both neg A as well as A prove the conjunction false:
(Atoneg{}B)toneg(Aland B)
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
(Ato(Bto C))to ( (Aland B)to C )
when C is a false proposition. Either of the above are constructively valid proofs by contradiction.Properties
commutativity: yes{| style="text-align: center; border: 1px solid darkgray;"|A land B| Leftrightarrow |B land A50px)| Leftrightarrow 50px)associativity: yes{| style="text-align: center; border: 1px solid darkgray;"|~A|~~~land~~~|(B land C)| Leftrightarrow |||(A land B)|~~~land~~~|~C50px)|~~~land~~~50px)| Leftrightarrow 50px)| Leftrightarrow 50px)|~~~land~~~50px)distributivity: with various operations, especially with or{| style="text-align: center; border: 1px solid darkgray;"|~A|land|(B lor C)| Leftrightarrow |||(A land B)|lor|(A land C)50px)|land50px)| Leftrightarrow 50px)| Leftrightarrow 50px)|lor50px){| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"! bgcolor="#ccccff"|others| with exclusive or:{| style="text-align: center; border: 1px solid darkgray;"|~A|land|(B oplus C)| Leftrightarrow |||(A land B)|oplus|(A land C)50px)|land50px)| Leftrightarrow 50px)| Leftrightarrow 50px)|oplus50px)with material nonimplication:{| style="text-align: center; border: 1px solid darkgray;"|~A|land|(B nrightarrow C)| Leftrightarrow |||(A land B)|nrightarrow|(A land C)50px)|land50px)| Leftrightarrow 50px)| Leftrightarrow 50px)|nrightarrow50px)with itself:{| style="text-align: center; border: 1px solid darkgray;"|~A|land|(B land C)| Leftrightarrow |||(A land B)|land|(A land C)50px)|land50px)| Leftrightarrow 50px)| Leftrightarrow 50px)|land50px)idempotency: yes{| style="text-align: center; border: 1px solid darkgray;"|~A~|~land~|~A~| Leftrightarrow |A~36px)|~land~36px)| Leftrightarrow 36px)monotonicity: yes{| style="text-align: center; border: 1px solid darkgray;"|A rightarrow B| Rightarrow |||(A land C)|rightarrow|(B land C)(File:Venn 1011 1011.svg|50px)| Rightarrow (File:Venn 1111 1011.svg|50px)| Leftrightarrow (File:Venn 0000 0101.svg|50px)|rightarrow(File:Venn 0000 0011.svg|50px)truth-preserving: yesWhen all inputs are true, the output is true.{| style="text-align: center; border: 1px solid darkgray;"|A land B| Rightarrow |A land B50px)| Rightarrow 60px)||(to be tested)}}falsehood-preserving: yesWhen all inputs are false, the output is false.{| style="text-align: center; border: 1px solid darkgray;"|A land B| Rightarrow |A lor B60px)| Rightarrow 50px)(to be tested)}}||Walsh spectrum: (1,-1,-1,1)Nonlinearity: 1 (the function is bent)If using binary values for true (1) and false (0), then logical conjunction works exactly like normal arithmetic multiplication.Applications in computer engineering{{anchor|software_AND}}
File:AND Gate diagram.svg|thumb|right|AND logic gatelogic gateIn high-level computer programming and digital electronics, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "AND", an algebraic multiplication, or the ampersand symbol "&". Many languages also provide short-circuit 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.
- 11000110 AND 10100011 = 10000010.
Set-theoretic correspondence
The membership of an element of an intersection set in set theory is defined in terms of a logical conjunction: x âˆˆ A âˆ© B if and only if (x âˆˆ A) âˆ§ (x âˆˆ B). Through this correspondence, set-theoretic intersection shares several properties with logical conjunction, such as associativity, commutativity, and idempotence.Natural language
As with other notions formalized in mathematical logic, the logical conjunction and is related to, but not the same as, the grammatical conjunction and in natural languages.English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order. For example, "They got married and had a child" in common discourse means that the marriage came before the child. 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 at once red, white, and blue, but rather that it has a part of each color.See also
{{colbegin|colwidth=20em}}- And-inverter graph
- AND gate
- Bitwise AND
- Boolean algebra (logic)
- Boolean algebra topics
- Boolean conjunctive query
- Boolean domain
- Boolean function
- Boolean-valued function
- Conjunction introduction
- Conjunction elimination
- De Morgan's laws
- First-order logic
- FrÃ©chet inequalities
- Grammatical conjunction
- Logical disjunction
- Logical negation
- Logical graph
- Logical value
- Operation
- Peanoâ€“Russell notation
- Propositional calculus
References
{{reflist}}External links
- {{springer|title=Conjunction|id=p/c025080}}
- Wolfram MathWorld: Conjunction
- WEB,weblink
- content above as imported from Wikipedia
- "Logical conjunction" does not exist on GetWiki (yet)
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- "Logical conjunction" does not exist on GetWiki (yet)
- time: 2:48pm EDT - Tue, May 21 2019
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