Do Intervals [2,3] and [2,5] in Real Numbers Share the Same Cardinality?

[FelixHelix]

Hi - I've got the following question but can't find any concrete information in my books on how to answer it and I'm slightly confused:

{x ε R : 2≤x≤3 } and {x ε R : 2≤x≤5 } Do they have the same cardinality?

My understanding of this is if you can find a mapping that satisifies a bijection then yes they do - but because the second set starts at 2 and not 4 I can't create this map and hence the second set will always be bigger. However they are both infinite so so do they share the cardinality ℂ?

Any ideas?

F

 

[micromass]

Hint: [0,1] and [0,2] have the same cardinality because the map

$$f:[0,1]\rightarrow [0,2]:x\rightarrow 2x$$

is a bijection.

Can you find a bijection between your two sets??

 

[FelixHelix]

Thanks. The only map I can see is (2^x) - x. I can't get from the second set back to the first. what is the preferred method to find this?

 

[micromass]

Thanks. The only map I can see is (2^x) - x. I can't get from the second set back to the first. what is the preferred method to find this?

That's good too, but not what I had in mind. If you want to get from the second set to the first: just find the inverse map.

Note that in this case, you can always find a map of the form $f(x)=ax+b$ for certain a and b.

 

[FelixHelix]

Ahh, I see. So y = 3x - 4 works!

 

[micromass]

Ahh, I see. So y = 3x - 4 works!

Indeed!

 

[alexfloo]

However they are both infinite so so do they share the cardinality ℂ?

Keep in mind that the integers and rational are both infinite also, but their cardinality is not c.