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Unformatted text preview: LECTURE 27: CARDINALITY OF POWER SETS ( 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 non-empty sets. We say that A has cardinality less than or equal to the cardinality of B , written | A | | B | if there is an injective function f : A B . We say A has cardinality strictly less than the cardinality of B , written | A | < | B | , if there is an injective function f : A B but there is no bijection between A and B . Example 2. We have | N | | Z | . In fact, | N | = | Z | . We also have | N | < | R | , since R is uncountable. There is a special notation for | N | ; it is written (read aleph-nought). 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? Weve 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