--- title: "Definitions as Design Decisions" description: "A definition specifies which objects a term includes. Comparing candidate definitions shows how their clauses affect equality, edge cases and the conclusions that follow." canonical: https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d source: https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d.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`. # Definitions as Design Decisions A definition specifies which objects a term includes. Comparing candidate definitions shows how their clauses affect equality, edge cases and the conclusions that follow. Subject: Mathematics · Area: Mathematical Thinking · Ages 22 to 24 Page: https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d ## Ready when they can - Compare two definitions of one idea and say what each one makes easy - Show that a theorem fails under a slightly weaker definition - Explain a definition's edge cases and why they are included or ruled out ## Lesson: Definitions are choices with a price You have checked proofs by naming their hypotheses. A definition supplies some of those hypotheses. A function's graph is its set of input-output pairs, with exactly one output for each domain input. Here, function equality means graph equality. Different formulas may describe the same graph. This specifies equality; it does not provide a finite algorithm for comparing every pair of infinite functions. **Example.** State domains before comparing formulas. On their largest real domains, h(x)=(x^2-4)/(x-2) excludes 2, while k(x)=x+2 accepts it. Factoring and cancelling shows equal outputs only for x≠2. The pair (2,4) belongs only to k's graph. Restrict both functions to the same inputs excluding 2, and their graphs are equal. *(drawing: The extra pair gives G an extra input. These finite graphs define different functions.)* For a declared domain A, every input needs exactly one output. Missing inputs violate totality; distinct outputs for one input violate uniqueness. Count extensions by multiplying the independent choices for missing inputs. A codomain B is a declared target containing the outputs. Some definitions make a map the triple (A,B,graph). Under that convention, changing B changes the map even if its graph stays fixed. Choosing requirements can determine an edge case. If 1! = 1 and the recurrence n! = n times (n-1)! applies at n=1, substitution forces 0! = 1. A different value breaks that recurrence. Alternatively, define n! as the number of permutations of an n-element set. A permutation is a bijection from a set to itself. The empty set has exactly one such bijection, the empty function, so this definition also gives 0! = 1. The empty product is 1 so that splitting a finite list into two disjoint lists preserves its product, even when one list is empty. Write E for the empty product. Splitting a list with product P into itself and an empty list requires P=P times E. Taking P=1 gives E=1. A real vector subspace W is a nonempty subset of a vector space, closed under addition and multiplication by every real scalar. Choose v in W; scalar closure gives 0 times v=0 in W. Nonemptiness permits that choice. The empty set passes both closure tests vacuously: it has no members that could fail them. Without nonemptiness, those closure conditions would not imply that W contains 0. **Example.** Test closure clauses separately. The integers are nonempty and closed under addition, but the real scalar 1/3 sends the integer 2 to 2/3, outside the set. The union of the coordinate axes in R^2 is nonempty and closed under real scalar multiplication. Addition fails: (2,0)+(0,3)=(2,3) lies on neither axis. Each example satisfies some clauses while exposing the job of another. A rational number is a ratio of integers with a nonzero denominator. Every integer k is rational because k=k/1. The number sqrt(2) is irrational. The rational indicator I(x) is 1 at rational inputs and 0 at irrational inputs. Classify each input, apply the rule, then add any requested values. For example, I(3/7)=1 and I(-sqrt(2))=0. Thomae's function T is 0 at irrational real inputs. At a rational input p/q in lowest terms, with q>0, it is 1/q. Reducing the fraction is part of the definition: 12/20=3/5 gives T(12/20)=1/5. At a rational input, 1/T(x) is the reduced positive denominator. An unreduced denominator would give different outputs for different expressions of the same number. Thomae's function is continuous at every irrational a. Fix ε>0. Only finitely many rationals in [a-1,a+1] have reduced positive denominator q≤1/ε: there are finitely many possible q and numerators. None equals a. Choose 0<δ<1 small enough to avoid them. Every remaining rational within δ has T<ε; irrationals have T=0. Since T(a)=0, this proves continuity. Also, sqrt(2)/2 is irrational: doubling a rational would be rational. Continuity at a means: for each ε>0, choose δ>0 so |x-a|<δ implies |f(x)-f(a)|<ε for domain inputs x. This δ may depend on a and ε. Uniform continuity means: for each ε>0, choose δ>0 so |x-y|<δ implies |f(x)-f(y)|<ε for every domain pair x,y. Here δ may depend on ε, but not on location. The output tolerance is ε; δ controls input distance. **Example.** For f(x)=x^2, continuity at a follows with δ=min(1,ε/(2|a|+1)): if |x-a|<δ, then |x+a|<2|a|+1 and |x^2-a^2|<ε. Uniform continuity fails on R. Fix ε=1. For any proposed δ>0, choose x=2/δ and y=x+δ/2. Their input gap is δ/2<δ, but y^2-x^2=2+δ^2/4>1. Such a failing pair for every δ disproves uniform continuity. A topology on X contains the empty set and X, and is closed under arbitrary unions and finite intersections. A family containing specified sets may need more members to satisfy these closure rules. On a finite X, build the smallest topology containing given sets by adding unions and finite intersections until nothing new appears. On X={1,2}, requiring {1} to be open gives just the empty set, {1} and X. **Recap.** Compare candidate definitions, probe their edge cases, and drop clauses to find which words do the work. *Lesson, questions and flashcards written for this dataset. Mathematics source records informed fact checks and exercise design; their passages and questions are not reproduced.* ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Writing a Proof Someone Else Can Check](https://lightmysky.com/learn/mathematics/writing-a-proof-someone-else-can-check-mt_nZ-zMzj18N) ## Optional connections - [Sigma-Algebras and Measurable Sets](https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk) - [Topological Spaces: Continuity Without Distance](https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI) ## Opens up - [Counterexamples and Testing a Conjecture](https://lightmysky.com/learn/mathematics/counterexamples-and-testing-a-conjecture-mt_SjA4jqNkaM)