**bag** Very loosely, a "set" that permits duplicates; more precisely, a collection of objects, called elements, in which the same element can appear any number of times. An example is the collection (*y,y,x,z,y,z*), which can equivalently be written as (*x,y,y,y,z,z*), since bags, like sets, have no ordering to their elements. The number of times a given element appears in a given bag is the multiplicity (of that element with respect to that bag).

The set theory operations of inclusion, union, intersection, difference, and product (but not complement) can all be generalized to apply to bags. First, inclusion: Let *b1* and *b2* be bags, and let element *x* appear exactly *n1* times in *b1* and exactly *n2* times in *b2* (*n1* 0, *n2* 0). Then bag *b1* includes ...

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