Post #1421423
2024-07-29 07:53 UTC
I helped a student make sense of direct proof, proof by contradiction and proof by contrapositive today, and it was quite successful, so I want to share it with you.
When you prove the statement “If A, then B” directly, your proof usually goes like this:
Suppose A
[insert arguments here]
Therefore B.
When you prove the statement “If A, then B” by contradiction, your proof usually goes like this:
Suppose A.
Suppose not B.
[insert arguments here]
C
But already, not C.
Contradiction!
Therefore B.
When you prove the statement “If A, then B” by contrapositive, your proof usually goes like this:
Suppose not B.
[insert arguments here]
Therefore not A.
Hence if A, then B.
It really helped the student to see how the two clauses in the original statement become sentences in the proof and where they go.
Replies (2)
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@KarenCampe@mathstodon.xyz 2024-07-29 12:41
@DavidKButler@mathstodon.xyz excellent!!
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@soaproot@sfba.social 2024-07-29 14:41
@DavidKButler@mathstodon.xyz Oooh nice. Also easily adapted to #constructiveMathematics as follows. The first and third work as you stated. The second one is the one which doesn't, but a modified variation of it does: When you prove the statement “If A, then not B” by contradiction (what @andrejbauer@mathstodon.xyz calls "proof of negation"), your proof usually goes like this: Suppose A. Suppose B. [insert arguments here] C But already, not C. Contradiction! Therefore not B.