Infinity

[shahar]

Between 1 and 2 there are infinity numbers and also between 1 to 1000 there are infinity numbers. Can I say that the infinity numbers between 1 to 1000 is "bigger" the infinity numbers between 1 to 2?

 

[pka]

NO! There is a one-to-one correspondence between $[1,2]~\&~[1,1000]$.
Can you find one?

 

[shahar]

NO! There is a one-to-one correspondence between $[1,2]~\&~[1,1000]$.
Can you find one?

O.K. I read about it by using function. But which function? How Can I show it by using a function?!

 

[pka]

There is no one function: i.e. there are several possible.
Here is one that I have used with classes.
Name points: $A:(1,1),~B:(2,1),~M:(1,0)~\&~N:(1000,0)$
Now the intervals $[1,2]~\&~[1,10^3]$ can be represented the line segments $\overline{AB}~\&~\overline{MN}~$.
The slope of the line $\ell:\overleftrightarrow {NB}$ is $\dfrac{-1}{998}$. Consider the point $\{P\}=\overleftrightarrow {AM}\cap\overleftrightarrow {NB}$
Take any point $T\in\overline{AB}$ and form $\overleftrightarrow {PT}$ That line must intersect $\overline {MN}$ in a unique point.
Do the same with $S\in\overline{MN}$ and form $\overleftrightarrow {PS}$ That line must intersect $\overline {AB}$ in a unique point.
Therefore we have a one-to-one correspondence between $\overline{AB}~\&~\overline {MN}$.

$$$$$$

 

[Deleted member 4993]

O.K. I read about it by using function. But which function? How Can I show it by using a function?!

Read "One, two, three, infinity..." by Asimov ----------------------------- edited (missed the three first time)

 

[topsquark]

Between 1 and 2 there are infinity numbers and also between 1 to 1000 there are infinity numbers. Can I say that the infinity numbers between 1 to 1000 is "bigger" the infinity numbers between 1 to 2?

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 $f: A \to B: a \mapsto a + 3$. 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 $f: A \to B: a \mapsto 100 a$. 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 $( -\infty, \infty )$. We call this cardinality $\aleph _1$. ( $\aleph _0$ is the cardinality of the integers.)

-Dan