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Additionally, a family of sets may be defined as a function from a set , known as the index set, to , in which case the sets of the family are indexed by members of . [1] In some contexts, a family of sets may be allowed to contain repeated copies of any given member, [ 2 ] [ 3 ] [ 4 ] and in other contexts it may form a proper class .
Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as math anxiety or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes ...
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing.
In fact, by de Morgan's laws, a δ-ring that contains is necessarily a σ-algebra as well. Fields of sets, and especially σ-algebras, are central to the modern theory of probability and the definition of measures. A semiring (of sets) is a family of sets with the properties
In fact, under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to . The first four properties listed above for multiplication say that Z {\displaystyle \mathbb {Z} } under multiplication is a commutative monoid .
Diophantus was born into a Greek family and is known to have lived in Alexandria, Egypt, during the Roman era, between AD 200 and 214 to 284 or 298. [6] [8] [9] [a] Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus.
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