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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 non-empty 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€śaleph-noughtâ€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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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11