Difference between revisions of "Set Law"

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A [[Set Law]] is an [[Addition]] that ...
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* <B>AKA:</B> [[Fundamentals]], [[Algebra of Sets]].
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* <B>See:</B> [[Axiomatic Set Theory]], [[Addition]], [[Multiplication]], [[Associativity]], [[Commutativity]], [[Reflexive Relation]], [[Antisymmetric Relation]], [[Transitive Relation]], [[Set (Mathematics)]], [[Naive Set Theory]], [[Axiom]].
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== References ==
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=== 2019 ===
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* (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Algebra_of_sets#Fundamentals Retrieved:2019-11-10.
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** The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic [[addition]] and [[multiplication]] are [[associativity|associative]] and [[commutativity|commutative]], so are set union and intersection; just as the arithmetic relation "less than or equal" is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]] and [[transitive relation|transitive]], so is the set relation of "subset". <P> It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on [[Set (mathematics)|sets]], for a fuller account see [[naive set theory]], and for a full rigorous [[axiom]]atic treatment see [[axiomatic set theory]].
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=== 2019 ===
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* (Wikipedia, 2019) &rArr; https://en.wikipedia.org/wiki/Algebra_of_sets#The_fundamental_properties_of_set_algebra Retrieved:2019-11-10.
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** The [[binary operation]]s of set [[Union (set theory)|union]] ( <math> \cup </math> ) and [[intersection (set theory)|intersection]] ( <math> \cap </math> ) satisfy many [[identity (mathematics)|identities]]. Several of these identities or "laws" have well established names. <P> :[[commutative operation|Commutative]] property: <P> ::* <math> A \cup B = B \cup A </math> ::* <math> A \cap B = B \cap A </math> :[[associativity|Associative]] property: <P> ::* <math> (A \cup B) \cup C = A \cup (B \cup C) </math> ::* <math> (A \cap B) \cap C = A \cap (B \cap C) </math> :[[distributivity|Distributive]] property: <P> ::* <math> A \cup (B \cap C) = (A \cup B) \cap (A \cup C) </math> ::* <math> A \cap (B \cup C) = (A \cap B) \cup (A \cap C) </math> The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection ''distributes'' over union. However, unlike addition and multiplication, union also distributes over intersection. <P> Two additional pairs of properties involve the special sets called the [[empty set]] Ø and the [[universe (mathematics)|universe set]] <math> U </math> ; together with the [[complement (set theory)|complement]] operator ( <math> A^C </math> denotes the complement of <math> A </math> . This can also be written as <math> A^' </math> , read as A prime). The empty set has no members, and the universe set has all possible members (in a particular context). <P> :Identity : <P> ::* <math> A \cup \varnothing = A </math> ::* <math> A \cap U = A </math> :Complement : <P> ::* <math> A \cup A^C = U </math> ::* <math> A \cap A^C = \varnothing </math> The identity expressions (together with the commutative expressions) say that, just like 0 and 1 for addition and multiplication, Ø and '''U''' are the [[identity element]]s for union and intersection, respectively. <P> Unlike addition and multiplication, union and intersection do not have [[inverse element]]s. However the complement laws give the fundamental properties of the somewhat inverse-like [[unary operation]] of set complementation. <P> The preceding five pairs of formulae&mdash;the commutative, associative, distributive, identity and complement formulae&mdash;encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them. <P> Note that if the complement formulae are weakened to the rule <math> (A^C)^C = A </math> , then this is exactly the algebra of propositional [[linear logic]].

Revision as of 23:03, 10 November 2019

A Set Law a Constraint in a Set System.


A Set Law is an Addition that ...



References

2019



2019

  • (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Algebra_of_sets#The_fundamental_properties_of_set_algebra Retrieved:2019-11-10.
    • The binary operations of set union ( [math] \cup [/math] ) and intersection ( [math] \cap [/math] ) satisfy many identities. Several of these identities or "laws" have well established names.

       :Commutative property:

       ::* [math] A \cup B = B \cup A [/math] ::* [math] A \cap B = B \cap A [/math] :Associative property:

       ::* [math] (A \cup B) \cup C = A \cup (B \cup C) [/math] ::* [math] (A \cap B) \cap C = A \cap (B \cap C) [/math] :Distributive property:

       ::* [math] A \cup (B \cap C) = (A \cup B) \cap (A \cup C) [/math] ::* [math] A \cap (B \cup C) = (A \cap B) \cup (A \cap C) [/math] The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over union. However, unlike addition and multiplication, union also distributes over intersection.

      Two additional pairs of properties involve the special sets called the empty set Ø and the universe set [math] U [/math] ; together with the complement operator ( [math] A^C [/math] denotes the complement of [math] A [/math] . This can also be written as [math] A^' [/math] , read as A prime). The empty set has no members, and the universe set has all possible members (in a particular context).

       :Identity :

       ::* [math] A \cup \varnothing = A [/math] ::* [math] A \cap U = A [/math] :Complement :

       ::* [math] A \cup A^C = U [/math] ::* [math] A \cap A^C = \varnothing [/math] The identity expressions (together with the commutative expressions) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively.

      Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.

      The preceding five pairs of formulae—the commutative, associative, distributive, identity and complement formulae—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.

      Note that if the complement formulae are weakened to the rule [math] (A^C)^C = A [/math] , then this is exactly the algebra of propositional linear logic.