Editing Distributive property (section)

{{Short description|Property involving two mathematical operations}}
{{redirect distinguish|Distributivity|Distributivism}}
{{Infobox mathematical statement
| name = Distributive property
| image = [[File:Illustration of distributive property with rectangles.svg|300px|class=skin-invert-image]]
| caption = Visualization of distributive law for positive numbers
| type = [[Principle|Law]], [[rule of replacement]]
| field = {{Plainlist|
* [[Elementary algebra]]
* [[Boolean algebra]]
* [[Abstract algebra]]
* [[Set theory]]
* [[Propositional calculus]]
}}
| statement = 
| symbolic statement = {{Plainlist|
# Elementary algebra
#: <math display="block">x \cdot (y + z) = x \cdot y + x \cdot z</math>
# Propositional calculus:
## <math>(P \land (Q \lor R)) \Leftrightarrow ((P \land Q) \lor (P \land R))</math>
## <math>(P \lor (Q \land R)) \Leftrightarrow ((P \lor Q) \land (P \lor R))</math>
}}
}}

In [[mathematics]], the '''distributive property''' of [[binary operation]]s is a generalization of the '''distributive law''', which asserts that the equality
<math display="block">x \cdot (y + z) = x \cdot y + x \cdot z</math>
is always true in [[elementary algebra]].
For example, in [[elementary arithmetic]], one has
<math display="block">2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3).</math>
Therefore, one would say that [[multiplication]] ''distributes'' over [[addition]].

This basic property of numbers is part of the definition of most [[algebraic structure]]s that have two operations called addition and multiplication, such as [[complex number]]s, [[polynomial]]s, [[Matrix (mathematics)|matrices]], [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]]. It is also encountered in [[Boolean algebra]] and [[mathematical logic]], where each of the [[logical and]] (denoted <math>\,\land\,</math>) and the [[logical or]] (denoted <math>\,\lor\,</math>) distributes over the other.