Solution 489Consider the relation on the set of transcendental numbers where
a∼b if and only if
a is a rational multiple of
b. Clearly such a relation is well-defined as the multiple of any transcendental number by an algebraic number is again a transcendental number.
It is clear that this relation is reflexive
a=1×a, and it is symmetric
a=nmb⇒b=mna, and furthermore, it is transitive
a=nmb,b=qpc⇒a=nqmpc. Thus the relation
∼ is an equivalence relation and partitions the set of transcendental numbers into disjoint sets.
Now the cardinality of each partition is countable due to the countability of the rationals, thus the number of partitions must be uncountable so as to satisfy the requirement that the cardinality of the set of transcendental numbers is uncountable.
Now if you consider the elements of the set
T/∼, none of them are rational multiples of one another, as if
a=nmb, then this means
a∼b, which is a contradiction as the partitions are disjoint.
Thus since none of the partitions are rational multiples of one another, they are definitely not integer multiples of one another, and thus we have shown that
S can be the size of the continuum.
S cannot be larger than the continuum, as that would imply that
R is larger than itself.