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The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A. There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class.
In set theory, a universal set is a set which contains all objects, including itself. [1] ... always has strictly higher cardinality than the set itself.
A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null , the smallest infinite cardinal In mathematics , a cardinal number , or cardinal for short, is what is commonly called the number of elements of a set .
Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also. As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the ...
A variant of set theory that includes a universal set and possibly other non-standard axioms, focusing on what can be constructed or defined positively. Polish space A Polish space is a separable topological space homeomorphic to a complete metric space pow Abbreviation for "power (set)" power "Power" is an archaic term for cardinality power ...
The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement, is a Σ-algebra over S and can be viewed as the prototypical example of a Boolean algebra.
The number of elements in a particular set is a property known as cardinality; informally, this is the size of a set. [5] In the above examples, the cardinality of the set A is 4, while the cardinality of set B and set C are both 3.
The principle of inclusion–exclusion, combined with De Morgan's law, can be used to count the cardinality of the intersection of sets as well. Let ¯ represent the complement of A k with respect to some universal set A such that for each k. Then we have