Distributive property

Distributive property

Visualization of distributive law for positive numbers
Type	Law, rule of replacement
Field	Elementary algebra Boolean algebra Abstract algebra Set theory Propositional calculus
Symbolic statement	Elementary algebra $${\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}$$ Propositional calculus: ${\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))}$ ${\displaystyle (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}$

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

This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted ${\displaystyle \,\land \,}$) and the logical or (denoted ${\displaystyle \,\lor \,}$) distributes over the other.