--- title: "Proof by Deduction and Exhaustion" description: "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." canonical: https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f source: https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Mathematical Thinking · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f ## Ready when they can - Prove that two arbitrary odd integers have an even sum by writing them as 2n + 1 and 2m + 1 - Prove a claim about the integers 1 to 5 by checking every case, and explain complete coverage in a case split - Explain why three confirming examples alone do not prove a claim for all integers ## Lesson: Small certain steps, every case checked Deduction uses stated assumptions, definitions and proved results, with a reason for each step. An even integer is twice an integer; an odd integer is 2n + 1 for some integer n. For integers n and m, (2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1). The bracket is an integer, so the sum is even. The two odd integers may differ, so n and m must be chosen independently. Using (2n + 1) + (2n + 1) covers equal odd integers only. To list positive odd integers, use n = 0, 1, 2, and so on. The values are 1, 3, 5, and so on. The rth positive odd integer uses n = r - 1. **Example.** Exhaustion checks every case in a stated finite domain. The notation n^2 means n times n. To prove n^2 + 3n is even for n in {1, 2, 3, 4, 5}, calculate each value: 4, 10, 18, 28 and 40. Each is even. List the allowed inputs and verify each calculation. A failed case disproves the claim. Inputs outside the stated domain are not required. *(drawing: Each cell pairs an allowed input with its calculated value. Every value is even.)* Three confirming integer checks alone do not prove a claim for all integers. Other inputs remain unchecked. A proof needs an argument covering every allowed input. One counterexample, an allowed input where the claim fails, disproves a claim about every input. To find one, calculate another case and compare the result with the claim. A case split must cover the whole domain and prove the claim in every group. Groups may overlap. General cases can cover infinitely many inputs. For any integer n, n^2 + n = n(n + 1). If n is even, n is an even factor. If n is odd, n + 1 is even. These cases cover every integer, so the product is always even. A proof needs correct calculations, even when an incorrect result happens to fit the claim. Check a factorisation by expanding it. For an inequality, calculate and compare both sides at each allowed input. The sign <= means less than or equal to, so equality passes that test. Assuming the conclusion and repeating it is circular reasoning; it does not prove the claim. **Recap.** Justify each deductive step. Exhaustion verifies every allowed case. Confirming examples alone do not prove an all-integer claim, but one counterexample disproves it. *Original lesson and practice questions written for this dataset.* ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Algebraic Proof with Odd, Even and Consecutive Numbers](https://lightmysky.com/learn/mathematics/algebraic-proof-with-odd-even-and-consecutive-numbers-mt_p9bajOC17q) ## Optional connections - [Proving Geometry Facts with Vectors](https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf) - [The Trapezium Rule](https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ) ## Opens up - [Loop Invariants and Proving an Algorithm Correct](https://lightmysky.com/learn/computing/loop-invariants-and-proving-an-algorithm-correct-mt_fWnstju2ns) - [Proof by Contradiction and Counterexample](https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8) - [Statements, Quantifiers and Negation](https://lightmysky.com/learn/mathematics/statements-quantifiers-and-negation-mt_S5NvyPxEZ_) - [Vector Spaces and Subspaces](https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n)