Information AboutCardinality |
| CATEGORIES ABOUT CARDINALITY | |
| cardinal numbers | |
| basic concepts in infinite set theory | |
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In Mathematics , the cardinality of a Set is a measure of the "number of Elements of the set". There are two approaches to cardinality – one which compares sets directly using Bijection s and Injection s, and another which uses Cardinal Number s. COMPARING SETS Two sets ''A'' and ''B'' have the same cardinality if there exists a Bijection , that is, an Injective and Surjective Function , from ''A'' to ''B''. For example, the set ''E'' = {2, 4, 6, ...} of Positive Even Number s has the same cardinality as the set N = {1, 2, 3, ...} of Natural Numbers , since the function ''f''(''n'') = 2''n'' is a bijection from N to ''E''. A set ''A'' has cardinality greater than or equal to the cardinality of ''B'' if there exists an injective function from ''B'' into ''A''. The set ''A'' has cardinality strictly greater than the cardinality of ''B'' if ''A'' has cardinality greater than or equal to the cardinality of ''B'', but ''A'' and ''B'' do not have the same cardinality. In other words, if there is an injective function from ''B'' to ''A'', but no bijective function from ''B'' to ''A''. For example, the set R of all Real Number s has cardinality strictly greater than the cardinality of the set '''N''' of all natural numbers, because the inclusion map ''i'' : '''N''' → R is injective, but it can be shown that there does not exist a bijective function from '''N''' to R. CARDINAL NUMBERS See Also: Cardinal number Above, "cardinality" was defined functionally. That is, the "cardinality" of a set was not defined as a specific object itself. However, such an object can be defined as follows. The relation of having the same cardinality is called Equinumerosity , and this is an Equivalence Relation on the Class of all sets. 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. The most common choice is the Initial Ordinal In That Class . This is usually taken as the definition of Cardinal Number in Axiomatic Set Theory . | ||
| Any Set ''X'' With Cardinality Less Than That Of The | "http://wwwinformationdelightinfo/information/entry/natural_number" class="copylinks">Natural Number s (''X'' < '''N''') is said to be a Finite Set |
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| Any Set ''X'' That Has The Same Cardinality As The Set Of The Natural Numbers (''X'' | '''N''' = <math>\aleph_0</math>) is said to be a Countably Infinite set |
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| Any Set ''X'' With Cardinality Greater Than That Of The Natural Numbers (''X'' > '''N''', For Example '''R''' | <math>\mathbf{c}</math> > '''N''') is said to be Uncountable |
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| If <code><math>X</math> | {a, b, c}</code> and <code><math>Y</math> = {apples, oranges, peaches}</code>, then <math>X = Y,</math> because <math>\{ \langle a, \mbox{apples}
angle, \langle b, \mbox{oranges}
angle, \langle c, \mbox{peaches}
angle \} </math> is a bijection between them Their cardinality is 3 |
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| If <math>X \leq Y</math>, Then There Exists <math>Z \,</math> Such That <math>X | Z\,</math>, and <math>Z \subseteq Y</math> |
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