42 - Comment The rational numbers are also countable In...

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Unformatted text preview: Comment The rational numbers are also countable. In contrast, Cantor showed that there are uncountable sets: that is, in- finite sets that are too large to be countable. An important example is the set of reals R . We cannot build up the reals in infinite stages, and this means that we cannot manipulate reals in the way we can natural numbers. Instead, we have to use approximations, such as the floating point deci- mals (given by the type Float in Haskell). Another example is the power set P ( N ) . For further information about infinite sets and countability see Truss, section 2.4. 5 Orderings Orderings are special relations which characterise when one object is ‘better’ than another. Orderings on sets of numbers such as N , Z and R are familiar: the ordering < describes the ‘less than’ ordering, and ≤ the ‘less than or equal’ ordering. We can also have orderings on other sets. For instance, suppose that we have a set of programs and we wish to distinguish which are cheaper, or run faster, or are more accurate.are cheaper, or run faster, or are more accurate....
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This note was uploaded on 03/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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