If the set is finite you count the number of distinct elements in it.
If the set has infinitely many elements, and you can find a one-to-one mapping between these elements and the natural numbers, then its cardinality is Aleph-null. Incidentally, the cardinality of rational numbers is also Aleph-null.
If you can map its elements to the set of real numbers, and if the continuum hypothesis is true then the cardinality of the set is the next transfinite number, Aleph-one. Unfortunately, if the Zermelo-Fraenkel set theory is consistent then neither the continuum hypothesis nor its negation can be proven. [It is not that nobody has proved it, but worse: as Godel proved, in any consistent and not-trivial mathematical theory, there are statements that cannot be proved to be true or false.]
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Deductive reasoning
what may be false
The only rule for any set is that given any element [number], you should be able to determine whether or not it is a member of the set.
You are using deductive logic.