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In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, [1] allows for multiple instances for each of its elements.The number of instances given for each element is called the multiplicity of that element in the multiset.
Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if and only if it is invertible; that is, a function : is bijective if and only if there is a function :, the inverse of f, such that each of the two ways for composing the ...
The set of all bags over type T is given by the expression bag T. If by multiset one considers equal items identical and simply counts them, then a multiset can be interpreted as a function from the input domain to the non-negative integers (natural numbers), generalizing the identification of a set with its indicator function. In some cases a ...
One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them.
In the prototypical example, one begins with a function : that takes two arguments, one from and one from , and produces objects in . The curried form of this function treats the first argument as a parameter, so as to create a family of functions f x : Y → Z . {\displaystyle f_{x}:Y\to Z.}
SSP can also be regarded as an optimization problem: find a subset whose sum is at most T, and subject to that, as close as possible to T. It is NP-hard, but there are several algorithms that can solve it reasonably quickly in practice. SSP is a special case of the knapsack problem and of the multiple subset sum problem.
A derangement of a set A is a bijection from A into itself that has no fixed points. Via the inclusion–exclusion principle one can show that if the cardinality of A is n, then the number of derangements is [n! / e] where [x] denotes the nearest integer to x; a detailed proof is available here and also see the examples section above.
In classical mathematics, characteristic functions of sets only take values 1 (members) or 0 (non-members). In fuzzy set theory, characteristic functions are generalized to take value in the real unit interval [0, 1], or more generally, in some algebra or structure (usually required to be at least a poset or lattice).