Prove that the open intervals 0 1 and 4 6 have the same card

Prove that the open intervals (0, 1) and (4, 6) have the same cardinality

Solution

Create an isomorphism f:R --> R such that f(0, 1) = (4, 6). The linear function...

f(x) = 4(1 - x) + 6x = (6 - 4)x + 4 =2x+4

...will do the trick. The map is injective since...

f(x) = f(y) <---> (6 - 4)x +4 = (6 - 4)y + 4 <---> x = y

...and it is surjective since, given any y in (a, b), we have...

f((y - a) / (b - a)) = y.

Thus, the intervals (0, 1) and (4, 6) have the same cardinality.