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These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
A subset V of A n is called an affine algebraic set if V = Z(S) for some S. [1]: 2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. [1]: 3 An irreducible affine algebraic set is also called an affine variety.
The external direct sum of a set of groups {H i} (written as Σ E {H i}) is the subset of Π{H i}, where, for each element g of Σ E {H i}, g i is the identity for all but a finite number of g i (equivalently, only a finite number of g i are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the ...
Given a family (repeats allowed) of subsets A 1, A 2, ..., A n of a universal set S, the principle of inclusion–exclusion calculates the number of elements of S in none of these subsets. A generalization of this concept would calculate the number of elements of S which appear in exactly some fixed m of these sets.
Given two sets A and B, A is a subset of B if every element of A is also an element of B. In particular, each set B is a subset of itself; a subset of B that is not equal to B is called a proper subset. If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A.
Throughout this article, capital letters (such as ,,,,, and ) will denote sets.On the left hand side of an identity, typically, will be the leftmost set, will be the middle set, and
A vector manifold is always a subset of Universal Geometric Algebra by definition and the elements can be manipulated algebraically. In contrast, a manifold is not a subset of any set other than itself, but the elements have no algebraic relation among them. The differential geometry of a manifold [3] can be
A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets [2] (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: [3]
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