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## Science Fair Project Encyclopedia for Schools!

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# Science Fair Project Encyclopedia

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# Collection

In common usage, a collection is any group of items that has one or more properties in common. For example, paintings from the same artist, or coins from Germany before 1900, see collecting and collectible.

In mathematics, collection is another term for set, or sometimes for class. Note that for sets in mathematics, repetition of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}. This is clearly different from ordinary usage of the term collection, for example a stamp collection may contain two copies of the same stamp. To model a set with possible multiple occurrence of elements, a function can be considered defined on the set of possible elements, of which the value is a non-negative integer denoting the multiplicity. Alternatively a function is considered defined on an index set, with values in the set of possible elements; in this case, to eliminate distinction regarding the order of elements, equivalence classes of such functions can be considered.

This concept of a set with possible multiple occurrence of elements is used implicitly in e.g. the fundamental theorem of algebra, which states: "a polynomial of degree n over the complex numbers has exactly n complex roots (not necessarily distinct ones)".

In music, specifically diatonic set theory, a collection is similar to a set, see generated collection.

03-10-2013 05:06:04
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