1. Sometimes to prove a statement false, we negate it and prove the negation is true. Is it also possible to prove a true statement by proving its negation is false with a counter example?
2. Could you explain proof by reduction?
Thanks in advance
More often, one derives a contradiction from the negation "For all x, P(x)" without exhibiting a counterexample explicitly. Sometimes this counterexample is still hidden in the proof, so it can be extracted and used to prove the original existential statement directly. Other times, extracting a counterexample cannot be done in principle.
No, in general a counterexample to the negation will not prove the statement. Mathematica theorems are typically of the form "for all x, P(x)". You could disprove that by giving a single counter-example, denying the "all x" part. But the negation of that is "for some x, not P(x)". A single counterexample, one value of x such that P(x) is true, does not disprove that.