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Distributive property

In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x ⋅ ( y + z ) = x ⋅ y + x ⋅ z {\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z} is always true in elementary algebra. For example, in elementary arithmetic, one has 2 ⋅ ( 1 + 3 ) = ( 2 ⋅ 1 ) + ( 2 ⋅ 3 ) . {\displaystyle…

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Field
Elementary algebra · Boolean algebra · Abstract algebra · Set theory · Propositional calculus
Symbolic statement
Elementary algebra x ⋅ ( y + z ) = x ⋅ y + x ⋅ z {\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z} · Propositional calculus: ( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) {\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))} ( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q… · ( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) {\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))} · ( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q ) ∧ ( P ∨ R ) ) {\displaystyle (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}
Type
Law, rule of replacement

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Overview

Definition

Meaning

Examples

Propositional logic

Distributivity and rounding

In rings and other structures

Generalizations

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Distributive property

Nodes100
Edges99
Triples19
Avg. degree1.98
Density0.02
Components1

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Distributive property

Top relations

Field · 5
Distributive property → Abstract algebra, Boolean algebra, Elementary algebra, Propositional calculus, Set theory
Symbolic statement · 4
Distributive property → ( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) {\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))}, ( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q ) ∧ ( P ∨ R ) ) {\displaystyle (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}, Elementary algebra x ⋅ ( y + z ) = x ⋅ y + x ⋅ z {\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}, Propositional calculus: ( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) {\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))} ( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q…
related to Distributivity and rounding · 3
Distributive property → For, In, Methods
related to Antidistributivity · 2
Distributive property → In, The
Type · 1
Distributive property → Law, rule of replacement

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Important terminology

displaystyle distributive multiplication addition law numbers operations distributivity algebra also property one cdot text two land lor operation distributes commutative

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Distributive propertyFieldElementary algebra1.00infobox
Distributive propertyFieldBoolean algebra1.00infobox
Distributive propertyFieldAbstract algebra1.00infobox
Distributive propertyFieldSet theory1.00infobox
Distributive propertyFieldPropositional calculus1.00infobox
Distributive propertySymbolic statementElementary algebra x ⋅ ( y + z ) = x ⋅ y + x ⋅ z {\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}1.00infobox
Distributive propertySymbolic statementPropositional calculus: ( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) {\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))} ( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q…1.00infobox
Distributive propertySymbolic statement( P ∧ ( Q ∨ R ) ) ⇔ ( ( P ∧ Q ) ∨ ( P ∧ R ) ) {\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))}1.00infobox
Distributive propertySymbolic statement( P ∨ ( Q ∧ R ) ) ⇔ ( ( P ∨ Q ) ∧ ( P ∨ R ) ) {\displaystyle (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}1.00infobox
Distributive propertyTypeLaw, rule of replacement1.00infobox
the algebra of sets or the switching algebra.Multiplying sums can be put into words as followsinstance ofExamples of structures with two operations that are each distributive over the other are Boolean algebras0.80text
banker's rounding may help in some casesinstance ofMethods0.80text

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