---
tag: 63JGSKKF7J
kind: concept
name: Distributive property
msc: [08A02]
sources:
  - url: https://en.wikipedia.org/wiki/Distributive_property
same_as:
  wikidata: Q187959
  wikipedia: Distributive property
---

## Summary

The distributive property links two [[binary operations|0M418FE7BB]] on one set: one of them can be applied term by term over the other.
The basic case is $a \cdot (b + c) = a \cdot b + a \cdot c$ for numbers, the [[distributive law|0KBEMAJAYA]].
Distributivity of multiplication over addition is an axiom of rings and fields, and distributivity of meet over join defines [[distributive lattices|0HM0HQCK6N]].

## Explanations

### L1 human

To compute $3 \cdot 12$, one can split $12$ as $10 + 2$ and compute $3 \cdot 10 + 3 \cdot 2 = 36$.
That this always works is the distributive property of multiplication over addition.
For sets, intersection distributes over union: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.
Addition does not distribute over multiplication: $1 + 2 \cdot 3 = 7$ but $(1 + 2) \cdot (1 + 3) = 12$.