cardinality - Cardinality The “cardinality” of a set...

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Unformatted text preview: Cardinality The “cardinality” of a set provides a precise meaning to “the number of elements” in the set, even when the set contains infinitely many elements. We start with the following very reasonable definitions of “the set S contains the same number of elements as the set T ”, “the set S contains fewer elements than the set T ” and “the set S contains more elements then the set T ”. Definition 1 Let S and T be nonempty sets. Then card ( S ) = card ( T ) ⇐⇒ there exists an f : S → T which is one–to–one and onto card ( S ) ≤ card ( T ) ⇐⇒ there exists an f : S → T which is one–to–one card ( S ) ≥ card ( T ) ⇐⇒ there exists an f : S → T which is onto card ( S ) < card ( T ) ⇐⇒ card ( S ) ≤ card ( T ) but card ( S ) 6 = card ( T ) card ( S ) > card ( T ) ⇐⇒ card ( S ) ≥ card ( T ) but card ( S ) 6 = card ( T ) Definition 2 Let S be a nonempty set. Then (a) The set S has card ( S ) = n ∈ IN if card ( S ) = card ( { 1 , 2 , 3 , ··· , n } ) . (b) The set S is countable if card ( S ) ≤ card (IN). The set S is countably infinite if card ( S ) = card (IN). (c) The set S is uncountable if it is not countable. (d) The set S has the cardinality of the continuum if card ( S ) = card (IR). We now verify that ≤ and ≥ , in the sense of cardinality, work as expected. Proposition 3 Let S , T and U be nonempty sets. Then (a) card ( S ) ≤ card ( T ) if and only if card ( T ) ≥ card ( S ) . (b) Either card ( S ) ≤ card ( T ) or card ( S ) ≥ card ( T ) . (c) If card ( S ) ≤ card ( T ) and card ( S ) ≥ card ( T ) , then card ( S ) = card ( T ) . (d) If card ( S ) ≤ card ( T ) and card ( T ) ≤ card ( U ) , then card ( S ) ≤ card ( U ) . Proof: (a) If card ( S ) ≤ card ( T ), then there is a one–to–one function f : S → T . Fix any s ∈ S ....
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This note was uploaded on 03/17/2010 for the course STATISTIC 472 taught by Professor Amjad during the Spring '08 term at Yarmouk University.

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cardinality - Cardinality The “cardinality” of a set...

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