Proof by Deduction and Exhaustion
Argue from what is given to what is claimed with every step justified, and settle a claim about a small finite set by checking every case in it.
What a learner can do afterwards
- Prove that the sum of two odd numbers is even by writing each as 2n + 1
- Prove a claim about the integers 1 to 5 by checking each case
- Explain why three worked examples do not prove a general statement
1 · Read
Deduction chains small certainties into big ones, and each link cites a definition, never a hunch. Write an odd number as 2n + 1, add two of them, and factor: (2n + 1) + (2m + 1) = 2(n + m + 1). The sum is twice an integer, which is exactly what even means.
Exhaustion works when the claim covers a small finite set you can meet face to face. To prove n squared + n is even for n = 1 to 5, check each door: 2, 6, 12, 20, 30, all even. List the set first so nothing hides, then verify every member.
Three worked examples prove nothing about all n, since infinitely many integers remain unchecked. A proof must handle an arbitrary n, using its definition rather than its value. If any sentence needs trust, split it finer until a sceptic can bite each link.
Splitting {1, 2, 3, 4, 5} into odds and evens can shorten exhaustion, but the blocks must rejoin to the whole. A case split that drops 5 proves nothing about 5. Cover everything, conquer each block, and confirm the cover.
Cite a definition for every deductive link, and check every case an exhaustion claim covers.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.