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Show that the intervals (0,1) and (-1,1) are equivalent and show that (-1,1) is equivalent to...

Show that the intervals (0,1) and (-1,1) are equivalent and show that (-1,1) is equivalent to the set of real numbers

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Answer #1

Each interval specifies a subset of real numbers.that is, the set of real numbers belonging to that interval.

If two sets are equal,only if they contain exactly same set of elements .

If two sets are equivalent, only if, the sets cardinality is same. That is, they must contain same number of elements.

Here the intervals, (0,1) and (-1,1) are countably infinite sets. According to cantor-Theorem ,the intervals cardinality is same.cardinality((0,1)) =cardinality((-1,1))=(read as Chi)

Similarly cardinality(set of real numbers R) =cardinality(-1,1)=

(since by Cartor theorem of countabilty infinite sets)

Therefore the R and (-1,1) are equivalent.

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