# Ring of sets

2010 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A05 [MSN][ZBL]

A collection $\mathcal{A}$ of subsets of a set $X$ satisfying:

i) $\emptyset\in \mathcal{A}$;

ii) $A\setminus B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$;

iii) $A\cup B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$.

It follows therefore that rings of sets are also closed under finite intersections. If the ring $\mathcal{A}$ contains $X$ then it is called an algebra of sets.

A $\sigma$-ring is a ring which is closed under countable unions, i.e. such that $\bigcup_{i=1}^\infty A_i \in \mathcal{A} \qquad \mbox{whenever } \{A_i\}_{i\in \mathbb N}\subset \mathcal{A}\, .$ A $\sigma$-ring is therefore closed under countable intersections. If the $\sigma$-ring contains $X$, then it is called a $\sigma$-algebra.

How to Cite This Entry:
Ring of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_of_sets&oldid=30105
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article