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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
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.
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.
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.
Three readings, ten practice questions and six flashcards written for this dataset. Mathematics source records informed fact checks and exercise design; their passages and questions are not reproduced.
More to read
Domains, graphs and codomains
A formula describes outputs, but a comparison of functions also needs their domains. Under graph equality, the complete set of input-output pairs determines a function. Removing an input removes its pair, even if the formula remains unchanged at every other input.
For a declared domain, exactly one output is required for every input. This combines totality and uniqueness. An unassigned input breaks totality. Two distinct outputs for one input break uniqueness. Different inputs may share an output. When completing a partial assignment, multiply the independent choices for the missing inputs.
The range consists of outputs actually attained. A codomain is a declared target containing those outputs. The graph records its range, but it does not record unused codomain elements. Some definitions therefore treat a map as a triple: domain, codomain and graph. Under that convention, changing the codomain changes the map. Under graph equality alone, it does not.
State the convention before using it. Both conventions require care when cancelling a denominator: the original domain restriction remains. Equality of graphs is a precise criterion, but it does not promise a finite procedure for comparing arbitrary infinite functions.
Empty cases and closure clauses
A definition can be chosen to preserve a useful identity. Requiring product splitting to work with an empty list determines the empty product. If E denotes that product, splitting a list with product P into itself and an empty list requires P=P times E. Testing a list with product 1 gives E=1.
The value of zero factorial has checkable consequences too. If 1! = 1 and n! = n times (n-1)! applies at n=1, then 0! = 1 follows. Under the counting definition, a permutation is a bijection from a set to itself. There is exactly one empty bijection, so the empty set has one permutation. Calling the starting requirements conventions does not prevent proving their consequences.
For a real vector subspace, nonemptiness and the two closure conditions do separate work. Nonemptiness supplies a vector v. Multiplication by real scalar 0 then puts the zero vector in the subset. The empty set passes the closure tests vacuously but supplies no v, showing why the nonemptiness clause is needed for this conclusion.
To test addition closure, seek two members whose sum leaves the subset. To test real scalar closure, seek a member and a real scalar whose product leaves it. One failing case is enough to disprove the corresponding universal requirement.
The order of choices in continuity
At a fixed point a, continuity allows an input distance δ to depend on both a and the requested output tolerance ε. Uniform continuity changes that dependence. Choose ε first, then choose δ, and use the same δ for every pair of domain inputs. Moving the choice of δ before the locations is a mathematical change, not a change in wording alone.
To disprove uniform continuity, fix one positive output tolerance. Then show that every proposed δ has a failing pair. For x^2 on the real line, take output tolerance 1. Given δ>0, choose x=2/δ and y=x+δ/2. The input gap is less than δ, but the squared outputs differ by 2+δ^2/4, greater than 1. No proposed distance works everywhere.
Thomae's definition uses a different restriction. Its value at an irrational is 0. At a rational p/q in lowest terms with q positive, its value is 1/q. Lowest terms matters because a number has many fractional expressions but needs one output.
At an irrational a, fix ε>0 and examine a bounded interval around a. Only finitely many rationals there have reduced denominators small enough to give values at least ε. None equals a. A sufficiently small neighbourhood avoids them all; every remaining value is below ε. This proves continuity at a. A finite sample of values alone would not prove the required statement about every nearby input.
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