Opposite of only once and only twice in theorems

Question

I want to know if something is said in a theorem like only once or even only twice and I want to use proof by contrapositive, how would we negate such statements. Basically, in terms of mathematics, what is the negation of only once. I think it is 0 or more times. If that is so, then what is the negation of only twice?

Answer 1

Negation of "only once" or "exactly once" is "less than once or greater than once". If it is only a count and cannot be less than, then the negation can be "zero or more than one time".

It is useful to think in terms of a variable, say $x$. Then negation of exactly once, i.e. $\neg(x = 1)$ is $(x < 1) \vee (x > 1)$.

Similarly, negation of exactly twice, i.e. $\neg(x = 2)$ is $(x < 2) \vee (x > 2)$.