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The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and a target-sum T {\displaystyle T} , and the question is to decide whether any subset of the integers sum to precisely T {\displaystyle T} . [ 1 ]
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 ...
If the base set is finite, then = ℘ since every subset of , and in particular every complement, is then finite.This case is sometimes excluded by definition or else called the improper filter on . [2] Allowing to be finite creates a single exception to the Fréchet filter’s being free and non-principal since a filter on a finite set cannot be free and a non-principal filter cannot contain ...
Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers , the set of rational numbers and the set of algebraic numbers are all countably infinite and therefore are null sets when considered as subsets of the real numbers.
As an illustration, let R be the set of real numbers, let Z be the set of integers, let O be the set of odd integers, and let P be the set of current or former U.S. Presidents. Then O is a subset of Z, Z is a subset of R, and (hence) O is a subset of R, where in all cases subset may even be read as proper subset. Not all sets are comparable in ...
For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N } forms a large sum-free subset of the set {1, ..., 2N }. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero n th powers of the integers is a sum-free set.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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