Reminder on ways to construct new sets out of old:

• set-difference -
• intersect ∩
• union ∪
• big-union (indexed) … index set!
  may not be integers:
  ∪_{{p \in PPG}} p.enemies() (where PPG is the set of all powerpuff-girls objects in my program)
  the union of a set-of-sets.
• power set
• cross product;
• (as a segue:)
  function spaces f: X → Y

Functions

Def'n: For sets X,Y, a (unary) function is an assignment from X (the ``domain'') to Y (the ``codomain'').

Examples of (unary) functions: length(•); car-color(•); colorname->rgb(•)

(partial functions: functions which throw an error on some inputs; often this is because our type system doesn't let us specify the input-set better. Formally, we might write ``f(17) = ⊥'' and add ⊥ to the codomain; may also represent inf. loop)

In java.util, there is a class which actually implements unary functions. Can you guess what it is?

• Exercise: describe each of the class's methods using standard function-terminology (``domain'', etc).
• Discuss: Are instances of these objects total functions, or partial? What about mutability — in the math world, values are values (you can't change the value of pi or of the function + or sqrt). Does this class implement mutable or immutable functions?

This is actually another method of constructing new sets from old: A → B is a set. Really, the ``:'' should be ``∈'' symbol.

It is also a type, in computer languages (which you write down in the function's contract). In programming languages, ``contract'' or ``type'' is sometimes also called ``signature'' (well, technically the signature includes the name of the function too).

What is the type of the function deriv? (ℜ→ℜ) → (ℜ→ℜ) Other functions in this set?

• ∫₇¹⁹
• the identity function;
• Given a stock over time, return a function: the historical doubling time: from when you invested $1, to how long until you have $2. Actually this function is ℜ→(ℜ ∪ ∞)

Def'n: a function f:A→A is idempotent if ∀ a ∈ A, f(f(a))=f(a).
Consider an installer program: what is its input? What does idempotency mean?

Building new functions from old

• Sum,product of functions:
  (f+g), (f*g), (fg).
  When do these make sense? Only when +,* defined on their domains (well, the cross-prod of their domains)
• Composing functions. Commutative? (What's the signature of ``compose''?)
• Conversely, Extending functions from elements onto sets:
  If A = {2,3,5}, then 4*A = {8,12,20}.
  Really, ``4*'' is just a function which happens to be defined on the elements of A. Consider also: A = {Barland,Cooper,Leebron}; numKids(A) = { 0, 2 }.
  In general, what do we mean by f(A)?
  We can extend f:A → B to a new function f^*: P(A) → P(B),
  where f^*(a^*) = { b | exists a ∈ a^* such that f(a)=b }
  Equivalently: f^*(a^*) = ∪_a∈a^* {f(a)}.
  [What is the code for this? (Btw, for list-represent, Cf. map)]
  What about binary functions? {2,3,5} ^ {-1,2} = {1/2,1/3,1/5, 4,9,25}.
  What is size of A^B ? Not |A|^|B|, certainly! Beware 2² vs 4¹?
  We can now phrase ``rationals closed under addition?'' as ``Q+Q ⊆? Q'' Is the set {Jan,Apr,Jul,Oct} closed under tax-refund-return-month?

More useful definitions:

• Image, f(S)
• codomain; the term ``range'' is deprecated these days.
• How might we define ``pre-image''?
• A set A is ``closed under f'' if f:A→A and f<sup>*</sup>(A) ⊆ A.
• def'n onto: (everything gets mapped to at least once)
  for each y ∈ Y, there is some x ∈ X s.t. f(x)=y
  Equivalently: f's image = f's entire co-domain.
• def'n 1-1, or into: (nothing is mapped to more than once)
  ∀x,y, f(x)=f(y) iff x=y.
• Inverse (def'n): A function f: X → Y is invertable iff:
  there exists some function g such that
  • for any x ∈ X, g(f(x)) = x [``g is a left-inverse of f''], and
  • for any y ∈ Y, f(g(y)) = y [``g is a right-inverse of f''].

Example: keybindings: Consider a program where it's possible to configure which key corresponds to which action (text editors, video games [ian: link to the appropriate lines of http://www.owlnet.rice.edu/~comp210/01fall/Assignments/Hw10/hw10-soln.ss, if public.] Discuss: Is this best described as a function, or as a relation? In each case, what is the domain/codomain? If a function: Is it 1-1, onto, invertible?
What about a keyboard-map (Qwerty, Dvorak; foreign-language glyphs)? What does it mean if a Xkeyboard map is not invertible?

1. 1. If f is invertable, then f is 1-1.
   2. If f is invertable, then f is onto.
2. If f is 1-1 and onto, then it's invertable.

1. 1. Suppose g is the inverse of f. We'll show f is 1-1:
      That is, we'll show: if f(x_a) = f(x_b), then x_a = x_b (for arbitray x_a, x_b).
      If f(x_a) = f(x_b), then g(f(x_a)) = g(f(x_b)) (apply g to equal things gives equal output).
      But because g is a (left) inverse, g(f(x_a)) = x_a and g(f(x_b)) = x_b,
      so we have that x_a = x_b, which completes the requirement for 1-1.
   2. Suppose g is the inverse of f. We'll show f is onto:
      That is, we'll show that for any y₀ ∈ Y, there is some x₀ ∈ X such that f(x₀) = y₀.
      Given an arbitrary y₀, which x₀ ∈ X will do the trick?
      g(y₀) will!, since f(g(y₀)) = y₀.
      [Note that g(y₀) is indeed an element of X, since g: Y → X.]
      [Interesting, a nice symmetry: this used g being a right-inverse of f.]
2. Suppose f is 1-1 and onto; we'll show that f has an inverse.
   

   We'll construct a function g:Y → X as follows, and then confirm that this particular g really does meet the requirements of ``inverse of f''.
   For any y₀ ∈ Y, we define g(y₀) as: the element x₀ such that f maps x₀ to y₀
   — such an x₀ exists because f is onto!

   Is this function g really an inverse of f? Let's look at the def'n of inverse:
   for any x ∈ X, what is g(f(x))? g(f(x)) is the element of X which f maps to f(x); this is x.
   So g(f(x)) = x, and our constructed function g is indeed a left-inverse to f.
   Similarly: for any y ∈ Y, what is f(g(y))? Well by our construction of g, g(y) is thing which f maps to y, so therefore f(g(y)) = y.

Woo-hoo! Um, wait a minute — in this second part, we used f being onto, but we didn't prove that f is 1-1. What gives? We'll use our debugging skills: Have we proven something stronger — that if f is onto, then it's invertable? If so, can we come up with a test-input? That is, make a function that is onto but not 1-1; does it have an inverse? Why not? What was the bug in our proof?

Upshot: In English, Beware the ``the''!
The word implies uniqueness and existence:

What a beautiful work of modern art — I'll marry the painter who created this!

• What if the work had been created by a team of several people [``the painter'' non-unique]
• What if a bucket had just been tipped over by the wind? [``the painter'' non-existent]
  [Unless of course the artist set up the bucket in a windy area, anticipating nature to complete their vision for them … or, they consider chance as ``art'', when you let it evoke an emotional reaction.]

For example, in our proof that 90=100, we glibly said ``let C be the point where two lines intersect''; technically we needed to add that such a point exists (easy to show the lines were non-parallel), and such a point is unique (clear, since two lines only intersect at most once, at least in Euclidean geometry.)

A few numeric functions:
⌊floor⌋, ⌈ceiling⌉; round-to-even; truncate-towards-0.

Reducing functions to sets: graphs!
Which is more fundamental: sets, or functions?
Between graphs and indicator functions: either!
[well okay — in a graph you need to lookup ]