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Countability (Mathematics) (view source)
Revision as of 22:38, January 6, 2010
, 22:38, January 6, 2010Created page with 'In mathematics, a set is called '''countable''' if it can be numbered in such a way that every element will eventually receive a number. ==Examples of Countable Sets== Obviously...'
In mathematics, a set is called '''countable''' if it can be numbered in such a way that every element will eventually receive a number.
==Examples of Countable Sets==
Obviously, the numbers 1, 2, 3, etc., are countable, but so are all the integers: 0 we call first, 1, second, -1, third, 2, fourth, -2, fifth, 3, sixth, and so on, going outwards. We're sure to hit every integer this way.
Somewhat surprisingly, the rational numbers are countable as well. See the picture below:
==Examples of Uncountable Sets==
The real numbers are not countable, nor is any set with positive [[Measure (Mathematics|Lebesgue measure]]. This is because any countable set of numbers can be completely contained within a set of arbitrarily small measure. A short proof follows:
==Examples of Countable Sets==
Obviously, the numbers 1, 2, 3, etc., are countable, but so are all the integers: 0 we call first, 1, second, -1, third, 2, fourth, -2, fifth, 3, sixth, and so on, going outwards. We're sure to hit every integer this way.
Somewhat surprisingly, the rational numbers are countable as well. See the picture below:
==Examples of Uncountable Sets==
The real numbers are not countable, nor is any set with positive [[Measure (Mathematics|Lebesgue measure]]. This is because any countable set of numbers can be completely contained within a set of arbitrarily small measure. A short proof follows: