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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 ...
where is an operator with two parameters—a one-parameter function, and a set to evaluate that function over. The other operators listed above can be expressed in similar ways; for example, the universal quantifier ∀ x ∈ S P ( x ) {\displaystyle \forall x\in S\ P(x)} can be thought of as an operator that evaluates to the logical ...
The only translation-invariant measure on = with domain ℘ that is finite on every compact subset of is the trivial set function ℘ [,] that is identically equal to (that is, it sends every to ) [6] However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover ...
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
That is, an enumeration of a set S is a bijective function from the natural numbers or an initial segment {1, ..., n} of the natural numbers to S. A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable.
A metric on a set X is a function (called the distance function or simply distance) d : X × X → R + (where R + is the set of non-negative real numbers). For all x, y, z in X, this function is required to satisfy the following conditions: d(x, y) ≥ 0 (non-negativity) d(x, y) = 0 if and only if x = y (identity of indiscernibles.
Every vector space is a free module, [1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S.
In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set. In mathematical analysis and in probability theory , the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection.