Sigma-ring
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High Quality Content by WIKIPEDIA articles! In mathematics, a nonempty collection of sets mathcal{R} is called a -ring (pronounced sigma-ring) if it is closed under countable union and relative complementation: 1. bigcup_{n=1}^{infty} A_{n} in mathcal{R} if A_{n} in mathcal{R} for all n in mathbb{N} 2. A - B in mathcal{R} if A, B in mathcal{R} From these two properties we immediately see that bigcap_{n=1}^{infty} A_n in mathcal{R} if A_{n} in mathcal{R} for all n in mathbb{N}This is simply because cap_{i=1}^infty A_n = A_1 - cup_{n=1}^{infty}(A_1-A_n). If the first property is weakened to…mehr

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High Quality Content by WIKIPEDIA articles! In mathematics, a nonempty collection of sets mathcal{R} is called a -ring (pronounced sigma-ring) if it is closed under countable union and relative complementation: 1. bigcup_{n=1}^{infty} A_{n} in mathcal{R} if A_{n} in mathcal{R} for all n in mathbb{N} 2. A - B in mathcal{R} if A, B in mathcal{R} From these two properties we immediately see that bigcap_{n=1}^{infty} A_n in mathcal{R} if A_{n} in mathcal{R} for all n in mathbb{N}This is simply because cap_{i=1}^infty A_n = A_1 - cup_{n=1}^{infty}(A_1-A_n). If the first property is weakened to closure under finite union (i.e., A cup B in mathcal{R} whenever A, B in mathcal{R}) but not countable union, then mathcal{R} is a ring but not a -ring.