Infinity
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pka
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There is no one function: i.e. there are several possible.
Here is one that I have used with classes.
Name points: [imath]A:(1,1),~B:(2,1),~M:(1,0)~\&~N:(1000,0)[/imath]
Now the intervals [imath][1,2]~\&~[1,10^3][/imath] can be represented the line segments [imath]\overline{AB}~\&~\overline{MN}~[/imath].
The slope of the line [imath]\ell:\overleftrightarrow {NB}[/imath] is [imath]\dfrac{-1}{998}[/imath]. Consider the point [imath]\{P\}=\overleftrightarrow {AM}\cap\overleftrightarrow {NB}[/imath]
Take any point [imath]T\in\overline{AB}[/imath] and form [imath]\overleftrightarrow {PT}[/imath] That line must intersect [imath]\overline {MN}[/imath] in a unique point.
Do the same with [imath]S\in\overline{MN}[/imath] and form [imath]\overleftrightarrow {PS}[/imath] That line must intersect [imath]\overline {AB}[/imath] in a unique point.
Therefore we have a one-to-one correspondence between [imath]\overline{AB}~\&~\overline {MN}[/imath].
[imath][/imath][imath][/imath][imath][/imath]
Here is one that I have used with classes.
Name points: [imath]A:(1,1),~B:(2,1),~M:(1,0)~\&~N:(1000,0)[/imath]
Now the intervals [imath][1,2]~\&~[1,10^3][/imath] can be represented the line segments [imath]\overline{AB}~\&~\overline{MN}~[/imath].
The slope of the line [imath]\ell:\overleftrightarrow {NB}[/imath] is [imath]\dfrac{-1}{998}[/imath]. Consider the point [imath]\{P\}=\overleftrightarrow {AM}\cap\overleftrightarrow {NB}[/imath]
Take any point [imath]T\in\overline{AB}[/imath] and form [imath]\overleftrightarrow {PT}[/imath] That line must intersect [imath]\overline {MN}[/imath] in a unique point.
Do the same with [imath]S\in\overline{MN}[/imath] and form [imath]\overleftrightarrow {PS}[/imath] That line must intersect [imath]\overline {AB}[/imath] in a unique point.
Therefore we have a one-to-one correspondence between [imath]\overline{AB}~\&~\overline {MN}[/imath].
[imath][/imath][imath][/imath][imath][/imath]
D
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Read "One, two, three, infinity..." by Asimov ----------------------------- edited (missed the three first time)
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topsquark
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The term "size" of a set refers to the "cardinality" of a set. Two sets have the same number of members (cardinality) if there is a bijection between them. So the set A = {1, 2, 3, 4} has the same cardinality as the set B = {4, 5, 6, 7} because there is a bijection [imath]f: A \to B: a \mapsto a + 3[/imath]. This is easy to see for finite sets but sets of infinite cardinality can be a bit tricky on the mind. The sets A = (0, 1) and B = (0, 100) have the same cardinality because we have the bijection [imath]f: A \to B: a \mapsto 100 a[/imath]. At first site it seems to be preposterous that the two sets should have the same number of members... shouldn't (0, 100) have 100 times more members than (0, 1)? Sure, it seems to make sense. But if you define cardinality to be the "size" of a set then the two have the same number of members.
It turns out that (0, 1) has the same cardinality as [imath]( -\infty, \infty )[/imath]. We call this cardinality [imath]\aleph _1[/imath]. ( [imath]\aleph _0[/imath] is the cardinality of the integers.)
-Dan