Proof by Contradiction Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Proof by contradiction is a powerful, widely used, but not always obvious way to prove certain claims. This discussion gives you some initial practice creating proofs by contradiction.

Consider the following claim:

Conjecture: For all integers n, if n ≡ 1 (mod 2), then n ≢ 2 (mod 4).

(If you see a congruence symbol with a slash right after it in this claim, that’s a “not congruent” symbol that some browsers apparently have trouble displaying correctly.)

How would you use proof by contradiction to prove this conjecture? For example, what would be the negation of the conjecture that you would try to prove? Where would you go from there? Do you either see a contradiction that follows from the negated conjecture, or a possible kind of contradiction that seems likely from it?