Definitions are choices with a price

Mathematics · Mathematical Thinking · ages 22-24

Name ______________________ Date ____________

Use each formula's largest real domain and graph equality. The formulas f(x)=x^2/x and g(x)=x define the same function.
Circle one: True False
A function has domain {a,b,c} and outputs in {0,1}. Its graph already contains (a,0) and (b,1). How many different full function graphs can extend these assignments?
Answer: ______________
For this question, a map is the triple (domain, codomain, graph). Two maps have the same domain and graph but different codomains. They are equal under this definition.
Circle one: True False
Under the convention that a function is its set of input-output pairs, what does this definition provide?
- A test for equality by comparing pair sets
- A shorter formula for every infinite function
- A fixed finite algorithm for evaluating every function
Thomae's function T is 0 at irrational inputs and 1/q at rational p/q in lowest terms, with q>0. What is 1/T(14/21)?
Answer: ______________
Let W={(s,0):s is real} union {(0,t):t is real}. Since W is nonempty and closed under multiplication by every real scalar, W is a real vector subspace of R^2.
Circle one: True False
The integers are a nonempty subset of the real vector space R and are closed under addition. Which fact shows why real scalar closure is also needed for a subspace?
- Multiplying 1 by 1/2 gives a number outside the integers
- Adding any two integers gives a number outside the integers
- The integers have no members, so nonemptiness is missing
- Multiplying an integer by another integer always leaves the integers
A real vector subspace is a nonempty subset closed under addition and real scalar multiplication. Sam says the empty set passes both closure tests vacuously. Is that claim correct?
- Yes. Both closure tests have no members to test
- No. The empty set contains vectors whose sum lies outside it
- No. Scalar multiplication sends a member outside the empty set
For a real-valued function on a real domain, which requirement defines uniform continuity rather than continuity at each point?
- For each ε, δ may depend on the particular pair being tested
- One fixed δ must work for every positive ε and every domain pair
- For each ε, one δ works for every domain pair
On X={a,b,c}, a family must contain the empty set, X, {a} and {b}. Definition A adds no rule; B also requires arbitrary unions and finite intersections. How many sets are in the smallest family satisfying B?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

Definitions are choices with a price W1-mt_b6ip8eP0-d-s1

False · The functions agree at every nonzero real input, but only g has the pair (0,0). Their graphs differ.
2 · Only c remains unassigned. It must receive exactly one of the two allowed outputs, giving two extensions.
False · The codomain is a component of the stated triple. Different codomains make these maps different under this convention.
A test for equality by comparing pair sets · Two functions are equal exactly when their full pair sets are equal under this convention. The definition does not promise a finite algorithm.
3 · Reduce 14/21 to 2/3. Then T(14/21)=1/3, whose reciprocal is 3.
False · Addition closure is missing. Both (1,0) and (0,1) lie in W, but their sum (1,1) lies on neither axis.
Multiplying 1 by 1/2 gives a number outside the integers · The real scalar 1/2 sends the integer 1 to 1/2, which is not an integer. Thus the integers fail real scalar closure.
Yes. Both closure tests have no members to test · There are no members that could fail either closure test. The empty set still fails the separate nonemptiness condition.
For each ε, one δ works for every domain pair · For each output tolerance ε>0, uniform continuity chooses δ>0 that works for all domain pairs. It may depend on ε, but not on location.
5 · B additionally forces {a,b}. The five sets empty set, {a}, {b}, {a,b} and X satisfy all its rules, so the smallest B-family has 5 members.