# Cardinality using equivalences

I proved that [0,1) has the same cardinality as (0,1], by defining a function and then checking injectivity/surjectivity.
I proved [0,1] has the same cardinality as (0,1), by defining a function and showing it has an inverse.
I now have to prove that (0,1] has the same cardinality as [0,1], and I can use any of the equivalences established above.

What method should I use to do this?

Edit: I know how to prove it using previous methods (defining a function and proving bijection), I just want to know if this can be done another way- using equivalence relations maybe?

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I proved that [0,1) has the same cardinality as (0,1], by defining a function and then checking injectivity/surjectivity.
I proved [0,1] has the same cardinality as (0,1), by defining a function and showing it has an inverse.
I now have to prove that (0,1] has the same cardinality as [0,1], and I can use any of the equivalences established above.

What method should I use to do this?

Edit: I know how to prove it using previous methods (defining a function and proving bijection), I just want to know if this can be done another way- using equivalence relations maybe?

I don't see any way of using equivalence relations.

|A|=|B| ⇔ There exists a bijection f:A→B

*where |A| denotes the cardinality of A*

So maybe you could say something like [0,1)U(0,1]=[0,1]

Then appeal to a theorem about the union of uncountably infinite sets?

Hope this helps