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Unformatted text preview: LECTURE 27: CARDINALITY OF POWER SETS ( Â§ 10.4) Most powerful is he who has himself in his own power. Seneca (5 BC  65 AD) 1. Comparing Cardinalities Intuitively, we know that  N  should be smaller than  R  . This is how can we make this precise: Definition 1. Let A and B be nonempty sets. We say that A has cardinality less than or equal to the cardinality of B , written  A  â‰¤  B  if there is an injective function f : A â†’ B . We say A has cardinality strictly less than the cardinality of B , written  A  <  B  , if there is an injective function f : A â†’ B but there is no bijection between A and B . Example 2. We have  N  â‰¤  Z  . In fact,  N  =  Z  . We also have  N  <  R  , since R is uncountable. There is a special notation for  N  ; it is written â„µ (read â€śalephnoughtâ€ť). Also, the cardinality of  R  is usually written c and called the continuum. The Continuum Question: Is there a cardinality between  N  and  R  ? Answer: Complicated! This was open for a long time. It depends on which axioms you assume! 2. How big is the power set? Weâ€™ve looked at direct products and their cardinalities. But how about the power set? Let A be a set. The power set P ( A ) is the set of all subsets of A ....
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 Spring '11
 Smith
 Sets, Finite set

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