66851

are equinumerous.

One can always find a one-to-one linear function of the form

f(x) = d-c b-ax + cb-ad

b-a transfonning all points of (a, b) into points of (c, d).

Theorem: Open interval (a, b) is eąuinumerous with the closed

interval [a, b|.

Let's dcfine in (a, b) a seąuence an = a + b-a 2n . The one-to-one

correspondence is given by the formula: f(a) = al; f(b) = a2; f(an) = an+2;

f(x) = x for others x ^ [a, b| (x 6= a, b, an).

Sets

Theorem: The open interval (a, b) is eąuinumerous with R (the

set of all real numbers).

Every open interval (a, b) one can transform on the interval

(-_

2, _

2 ) and then, function f(x) = tan x (which is one-to-one in (-_

2, _

2 )) transforms the interval (-_

2 , _