That's when you say two or more things that cannot both be true, such as, "The one and only solution is 3," and, "The one and only solution is 17."
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There is no contradiction.
Proof by contradiction is also known by its Latin equivalent, reductio ad absurdum.
true
Yes. A number can be either rational or irrational, but never both; otherwise there would be an inherent contradiction.
Yes, that's how it is done. Assuming the contrary should eventually lead you to some contradiction.