# Cardinality and Equivalence

1. Feb 7, 2012

### 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

2. Feb 7, 2012

### 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??

3. Feb 7, 2012

### 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 prefered method to find this?

4. Feb 7, 2012

### micromass

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.

5. Feb 7, 2012

### FelixHelix

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

6. Feb 7, 2012

Indeed!!

7. Feb 7, 2012

### alexfloo

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

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