distributivity Distributive 2003-07-22 13:10:59 2007-06-29 04:04:39 Definition distributive left distributive right distributive left-distributive right-distributive distributes over left distributivity right distributivity left distributes over left distributive law right distributive law \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} Given a \PMlinkname{set}{Set} $S$ with two binary operations $+\colon S \times S \to S$ and $\cdot\colon S \times S \to S$, we say that $\cdot$ is {\em right distributive} over $+$ if $$(a+b) \cdot c = (a \cdot c) + (b \cdot c)\mathrm{~for~all~} a,b,c \in S$$ and {\em left distributive} over $+$ if $$a \cdot (b+c) = (a \cdot b) + (a \cdot c)\mathrm{~for~all~}a,b,c \in S.$$ If $\cdot$ is both left and right distributive over $+$, then it is said to be {\em distributive} over $+$ (or, alternatively, we may say that $\cdot$ {\em distributes over} $+$).