11Two sets are considered equinumerous if they have the same number of elements. for
example, the set {2 5 6} is equinumerous with the set {-512 5 345}. Also, the set of all
natural numbers {1 2 3 4 5...} has the same number of elements as the set of all even
positive integers {2 4 6 8 10...}; they both have a cardinality of aleph null, meaning each
has aleph null elements. To prove that sets with a transfinite number of elements have
the same number of elements, a bijective, or one-to-one function from one set to the
other must exist. So, we can prove that there are as many even postive integers as there
are natural numbers by finding the bijective function between the two, and that function
is f(x) = 2x.
So, to prove the statement why is the total number of real numbers equal to the
number of numbers between 0 n 1?? , we have to find a bijective function mapping
the numbers in the interval (0,1) onto the set of all real numbers, from negative infinity to
positive infinity. The first such function I could find was a piecewise function defined as
f(x) = (ln(x)-ln(1/2)) when x is less than 1/2 and f(x) = -ln(1-x) + ln(1/2) when x is
greater than 1/2. Since a bijective function between (0,1) and the set of all real numbers
exists, the two sets are equinumerous, and each contains the same number of elements.
