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Irrational numbers 
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#19
Jun1414, 12:35 PM

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P: 2,097

Two sets [itex]A[/itex] and [itex]B[/itex] are said to be "the same size" (technically, the same cardinality) if you can set up a correspondence between the two sets, so that every element of [itex]A[/itex] is matched with exactly one element of [itex]B[/itex], and viceverse (technically, a onetoone mapping). For example, the sets [itex]A = \{ cat, dog, pig \}[/itex] [itex]B = \{ red, yellow, blue\}[/itex] are the same size because they can be put into correspondence many different ways, but here's one: [itex]cat \leftrightarrow red,\ dog \leftrightarrow yellow,\ pig \leftrightarrow blue[/itex] Infinite sets can be put into a onetoone correspondence, also. For example, the set [itex]A = [/itex] the positive integers and the set [itex]B = [/itex] all integers: [itex]1 \leftrightarrow 0[/itex] [itex]2 \leftrightarrow 1[/itex] [itex]3 \leftrightarrow +1[/itex] [itex]4 \leftrightarrow 2[/itex] [itex]5 \leftrightarrow +2[/itex] etc. You can also set up a onetoone correspondence between the integers and the rationals. That's a little harder to describe, but it can be done. Any set that can be put into a onetoone correspondence with the positive integers is called a "countable" set. Some sets are not countable. The easiest example is the set of reals. There is no way to set up a onetoone correspondence between the positive integers and the reals. 


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