In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets is called a ring (of sets) if it is closed under intersection and union. That is, for any ,
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In measure theory, a ring of sets is instead a family closed under unions and set-theoretic differences. That is, it obeys the two properties
- and
This implies that it is also closed under intersections, because of the identity
however, a family of sets that is closed under unions and intersections might not be closed under differences.
Read more about Ring Of Sets: Examples, Related Structures
Famous quotes containing the words ring of, ring and/or sets:
“I was exceedingly interested by this phenomenon, and already felt paid for my journey. It could hardly have thrilled me more if it had taken the form of letters, or of the human face. If I had met with this ring of light while groping in this forest alone, away from any fire, I should have been still more surprised. I little thought that there was such a light shining in the darkness of the wilderness for me.”
—Henry David Thoreau (1817–1862)
“He will not idly dance at his work who has wood to cut and cord before nightfall in the short days of winter; but every stroke will be husbanded, and ring soberly through the wood; and so will the strokes of that scholar’s pen, which at evening record the story of the day, ring soberly, yet cheerily, on the ear of the reader, long after the echoes of his axe have died away.”
—Henry David Thoreau (1817–1862)
“bars of that strange speech
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And ends as one grand transcendental vowel.”
—Randall Jarrell (1914–1965)