define a bijection between 0 1 0 1 0 1 and 0 1SolutionWe k

define a bijection between [0, 1] × [0, 1] × [0, 1] and [0, 1].

Solution

We know that [0,1] is bijective with the set of all binary sequences of arbitrary length.

So any element of the LHS can be identified with

(a,b,c) where a, b, c are sequences a[k],b[k],c[k] taking values in 0 and 1 .

So we have a map (a,b,c) to the sequence a[1]b[1]c[1]a[2]b[2]c[2]..........

We can modify this to make one to one.

This proves that cardinality of [0,1]x[0,1]x[0,1] is not bigger than that of [0,1].

The other way is obvious --we can always use the diagonal embedding.

Hence the claim